Total Dual Integrality of Triangle Covering

Abstract

This paper concerns weighted triangle covering in undirected graph $G=(V,E)$, where a nonnegative integral vector $\mathbf w=(w(e):e\in E)^T$ gives weights of edges. A subset S of E is a triangle cover in G if S intersects every triangle of G. The weight of a triangle cover is the sum of w(e) over all edges e in it. The characteristic vector $\mathbf x$ of each triangle cover in G is an integral solution of the linear system

$$\begin{aligned} \pi :A\mathbf x\ge \mathbf 1,\mathbf x\ge \mathbf 0, \end{aligned}$$

where A is the triangle-edge incidence matrix of G. System $\pi$ is totally dual integral if $\max \{\mathbf 1^T\mathbf y:A^{T}\mathbf y\le \mathbf w,\mathbf y\ge \mathbf 0\}$ has an integral optimum solution $\mathbf y$ for each integral vector $\mathbf w\in \mathbb Z_+^E$ for which the maximum is finite. The total dual integrality of $\pi$ implies the nice combinatorial min-max relation that the minimum weight of a triangle cover equals the maximize size of a triangle packing, i.e., a collection of triangles in G (repetitions allowed) such that each edge e is contained in at most w(e) of them. In this paper, we obtain graphical properties that are necessary for the total dual integrality of system $\pi$, as well as those for the (stronger) total unimodularity of matrix A and the (weaker) integrality of polyhedron $\{\mathbf x:A\mathbf x\ge \mathbf 1,\mathbf x\ge \mathbf 0\}$. These necessary conditions are shown to be sufficient when restricted to planar graphs. We prove that the three notions of integrality coincide, and are commonly characterized by excluding odd pseudo-wheels from the planar graphs.

Research supported in part by NNSF of China under Grant No. 11531014 and 11222109.

1 Introduction

Covering and packing triangles in graphs has been extensively studied for decades in graph theory [6, 7, 14] and optimization theory [2, 9]. In this paper, we study the problem from both a polyhedral perspective and a graphical persective – characterizing polyhedral integralities of triangle covering and packing with graphical structures.

Graphs considered in this paper are undirected, simple and finite. A weighted graph $(G,\mathbf w)$ consists of a graph G (with vertex set V(G) and edge set E(G)) and an edge weight (function) $\mathbf w\in \mathbb Z_+^{E(G)}$. The weight of any edge subset S is $w(S)=\sum _{e\in S}w(e)$. By a triangle cover of G we mean an edge subset S ($\subseteq E(G)$) whose removal from G leaves a triangle-free graph. Let $\tau _w(G)$ denote the minimum weight of a triangle cover of $(G,\mathbf w)$. By a triangle packing of $(G,\mathbf w)$ we mean a collection of triangles in G (repetition allowed) such that each edge $e\in E(G)$ is contained in at most w(e) of them. Let $\nu _w(G)$ denote the maximum size of a triangle packing of $(G,\mathbf w)$. In case of $\mathbf w=\mathbf 1$, we write $\tau _w(G)$ and $\nu _w(G)$ as $\tau (G)$ and $\nu (G)$, respectively.

Tuza’s Conjecture and Variants. A vast literature on triangle covering and packing concerns Tuza’s conjecture [14] that $\tau (G)\le 2\nu (G)$ for all graphs G and its weighted version [2] that $\tau _w(G)\le 2\nu _w(G)$ for all graphs G and all $\mathbf w\in \mathbb Z_+^{E(G)}$. Both conjectures remain wide open. The best known general results $\tau (G)\le 2.87\nu (G)$ and $\tau _w(G)\le 2.92\nu _w(G)$ are due to Haxell [7] and Chapuy et al. [2], respectively. Many researchers have pursued the conjectures by showing the conjectured inequalities hold for certain special class of graphs. In particular, Tuza [15] and Chapuy et al. [2] confirmed their own conjectures for planar graphs. Haxell et al. [6] proved the stronger inequality $\tau (G)\le 1.5\nu (G)$ if G is planar and $K_4$-free, where $K_4$ denotes the complete graph on 4 vertices.

Along a different line, Lakshmanan et al. [10] proved that the equation $\tau (G)=\nu (G)$ holds whenever G is ($K_4$, gem)-free or G’s triangle graph is odd-hole-free. A natural question arises for the weighted version: When does $\tau _w(G)=\nu _w(G)$ hold? This question is closely related to the notion of total dual integrality from the theory of polyhedral combinatorics.

Total Dual Integrality. A rational system $\{A\mathbf x\ge \mathbf b,\mathbf x\ge \mathbf 0\}$ is called totally dual integral (TDI) if the maximum in the LP duality equation

$$\begin{aligned} \min \{\mathbf c^T\mathbf x:A\mathbf x\ge \mathbf b,\mathbf x\ge \mathbf 0\}=\max \{\mathbf b^T\mathbf y:A^{T}\mathbf y\le \mathbf c,\mathbf y\ge \mathbf 0\} \end{aligned}$$

has an integral optimum solution $\mathbf y$ for each integral vector $\mathbf c$ for which the maximum is finite. The model of TDI systems introduced by Edmonds and Galies [5] plays a crucial role in combinatorial optimization and serves as a general framework for establishing many important combinatorial min-max relations [3, 4, 11, 12]. Schrijver and Seymour [13] derived the following useful tool for proving total dual integrality.

Theorem 1

[13]. The rational system $A\mathbf x\ge \mathbf b,\mathbf x\ge \mathbf 0$ is TDI, if and only if

$$\begin{aligned} \max \{\mathbf b^T\mathbf y:A^{T}\mathbf y\le \mathbf c,\mathbf y\ge \mathbf 0,2\mathbf y\,{ is\, integral}\} \end{aligned}$$

has an integral optimum solution $\mathbf y$ for each integral vector $\mathbf c$ for which the maximum is finite.

Edmonds and Giles [5] showed that total dual integrality implies primal integrality as specified by the following theorem.

Theorem 2

[5]. If rational system $A\mathbf x\ge \mathbf b,\mathbf x\ge \mathbf 0$ is TDI and $\mathbf b$ is integral, then the polyhedron $\{\mathbf x: A\mathbf x\ge \mathbf b,\mathbf x\ge \mathbf 0\}$ is integral, i.e., $\min \{\mathbf c^T\mathbf x:A\mathbf x\ge \mathbf b,\mathbf x\ge \mathbf 0\}$ is attained by an integral vector for each integral vector $\mathbf c$ for which the minimum is finite.

Given a weighted graph $(G,\mathbf w)$, let $\varLambda (G)$ denote the set of triangles in G. To see the relation between the equation $\tau _w(G)=\nu _w(G)$ and TDI systems, let us consider the hypergraph $\mathcal H_G=(E(G),\varLambda (G))$ of triangles in G. We assume $\varLambda (G)\ne \emptyset$ to avoid triviality. The edge-vertex incidence matrix $A_G$ of $\mathcal H_G$ is exactly the triangle-edge incidence matrix of G, whose rows and columns are indexed by triangles and edges of G, respectively, such that for any $\triangle \in \varLambda (G)$ and $e\in E(G)$, $A_{\triangle ,e}=1$ if $e\in \triangle$ and $A_{\triangle ,e}=0$ otherwise. In standard terminologies from the theory of packing and covering [4, 12], we write

$$\begin{aligned} \tau _w(\mathcal H_G)= & {} \min \{\mathbf w^T\mathbf x:A_G\mathbf x\ge \mathbf 1,\mathbf x\in \mathbb Z_+^{E(G)}\}, \end{aligned}$$

(1.1)

$$\begin{aligned} \nu _w(\mathcal H_G)= & {} \max \{\mathbf 1^T\mathbf y:A^T_G\mathbf y\le \mathbf w,\mathbf y\in \mathbb Z_+^{\varLambda (G)}\}, \end{aligned}$$

(1.2)

$$\begin{aligned} \tau ^*_w(\mathcal H_G)= & {} \min \{\mathbf w^T\mathbf x:A_G\mathbf x\ge \mathbf 1,\mathbf x\ge \mathbf 0\}, \end{aligned}$$

(1.3)

$$\begin{aligned} \nu ^*_w(\mathcal H_G)= & {} \max \{\mathbf 1^T\mathbf y:A^T_G\mathbf y\le \mathbf w,\mathbf y\ge \mathbf 0\}. \end{aligned}$$

(1.4)

Combinatorially, each feasible 0–1 solution $\mathbf x$ of (1.1) is the characteristic vector of a triangle cover of G, and vice versa. Thus such an $\mathbf x$ is also referred to as a triangle cover (or an integral triangle cover to emphasis the integrality) of G. Moreover the minimality of $\tau _w(\mathcal H_G)$ implies that

$$\begin{aligned} \tau _w(\mathcal H_G)=\tau _w(G). \end{aligned}$$

Similarly, each feasible solution $\mathbf y$ of (1.2) is regarded as a triangle packing (or an integral triangle packing) which contains, for each $\triangle \in \varLambda (G)$, exactly $y(\triangle )$ copies of $\triangle$. In particular,

$$\begin{aligned} \nu _w(\mathcal H_G)=\nu _w(G). \end{aligned}$$

Usually, feasible solutions of (1.3) and (1.4) are called fractional triangle covers and fractional triangle packings of G, respectively. Writing $\tau ^*_w(G)=\tau ^*_w(\mathcal H_G)$ and $\nu ^*_w(G)=\nu ^*_w(\mathcal H_G)$, the LP-duality theorem gives

$$\begin{aligned} \tau _w(G)\ge \tau ^{*}_w(G)= \nu ^{*}_w(G)\ge \nu _w(G). \end{aligned}$$

It is well known (see e.g., page 1397 of [12]) that

$$\begin{aligned} \tau _w(G)= \nu _w(G)\,\text {holds for each}\, \mathbf w\in \mathbb Z_{+}^{E(G)}\,\text {if and only if}\,A_G\mathbf x\ge \mathbf 1,\mathbf x\ge \mathbf 0\,\text {is TDI}. \end{aligned}$$

Total Unimodularity. A matrix A is totally unimodular (TUM) if each subdeterminant of A is 0, 1 or $-1$. Total unimodular matrices often imply stronger integrality than TDI systems (see e.g., [8]).

Theorem 3

An integral matrix A is totally unimodular if and only if the system $A\mathbf x\ge \mathbf b,\mathbf x\ge \mathbf 0$ is TDI for each vector $\mathbf b$.

The 0–1 TUM matrices are connected to balanced hypergraphs. Let $\mathcal H=(\mathcal V,\mathcal E)$ be a hypergraph with vertex set $\mathcal V$ and edge set $\mathcal E$. Let $k\ge 2$ be an integer. In $\mathcal H$, a cycle of length k is a sequence $v_{1}e_{1}v_{2}e_{2}\ldots v_{k}e_{k}v_{1}$ such that $v_{1}, \ldots , v_{k} \in \mathcal V$ are distinct, $e_{1},\ldots , e_{k}\in \mathcal E$ are distinct, and $\{v_{i},v_{i+1}\}\subseteq e_{i}$ for each $i=1,\ldots ,k$, where $v_{k+1}=v_{1}$. Hypergraph $\mathcal H$ is called balanced if every odd cycle, i.e., cycle of odd length, has an edge that contains at least three vertices of the cycle.

Theorem 4

(Berge [1]). Let $\mathcal H$ be a hypergraph such that every edge consists of at most three vertices. Then the vertex-edge incidence matrix of $\mathcal H$ is TUM if and only if $\mathcal H$ is balanced.

Our Results. Let $\mathfrak B$, $\mathfrak M$, and $\mathfrak I$ be the sets of graphs G such that the triangle-edge incidence matrices $A_G$ are TUM, systems $A_G\mathbf x\ge \mathbf 1,\mathbf x\ge \mathbf 0$ are TDI, and polyhedra $\{\mathbf x|A_G\mathbf x\ge \mathbf 1,\mathbf x\ge \mathbf 0\}$ are integral, respectively. In terminologies of hypergraph theory (see e.g., Part VIII of [12]),

$$\begin{aligned} G\in \mathfrak B\,\Leftrightarrow & {} \mathcal H_G\text { is balanced (by Theorem}\, 4 \,\text {because}\, \mathcal H_G \,\text {is 3-uniform)}.\\ G\in \mathfrak M\Leftrightarrow & {} \mathcal H_G\text { is Mengerian, i.e., }\tau _w(G)= \nu _w(G)\text { holds for each }\mathbf w\in \mathbb Z_{+}^{E(G)}.\\ G\in \,\mathfrak I\,\,\Leftrightarrow & {} \mathcal H_G\text { is ideal, i.e., }\tau _w(G)= \tau ^{*}_w(G)\text { holds for each }\mathbf w\in \mathbb Z_{+}^{E(G)}. \end{aligned}$$

Recalling Theorems 2 and 3, given any graph G, the total modularity (balanced-ness): $G\in \mathfrak B$ implies the total dual integrality (Mengerian property): $G\in \mathfrak M$, while $G\in \mathfrak M$ implies primal integrality: $G\in \mathfrak I$. It follows that

$$\begin{aligned} \mathfrak B\subseteq \mathfrak M\subseteq \mathfrak I. \end{aligned}$$

(1.5)

In Sect. 2, first we strengthen (1.5) to $\mathfrak B\varsubsetneq \mathfrak M\varsubsetneq \mathfrak I$ (Theorem 5). Then we obtain necessary conditions for a graph to be a member of $\mathfrak I$ (Lemma 4) or a minimal graph outside $\mathfrak B$ (Theorem 6 and its corollaries) in terms of the pattern of the so-called odd triangle-cycles (Definition 1). Building on these conditions, we establish in Sect. 3 the following characterization for total dual integrality of covering triangle in planar graphs G (Theorem 9):

$$ \begin{aligned} {G} \in \mathfrak M\Leftrightarrow G\in \mathfrak B\Leftrightarrow G\in \mathfrak I\,\text {is}\, K_4\text {-free}\,\Leftrightarrow G\,\text {is}\, K_4\text {-free}\, \& \,\text {odd pseudo-wheel-free}, \end{aligned}$$

where odd pseudo-wheels correspond to odd induced cycles in the triangle graph of G (Definition 2). We conclude in Sect. 4 with remarks on characterizing general graphs $G\in \mathfrak M$ and general graphs $G\in \mathfrak I$. For easy reference, Appendix gives a list of mathematical symbols used in the paper.

2 General Graphs

In this section, we study TUM, TDI and integral properties for covering and packing triangle in general graphs. We often identify a graph G with its edge set E(G). The following definition is crucial to our discussions.

Definition 1

A triangle-cycle in G is a sequence $C=e_1\triangle _1e_2\cdots e_k\triangle _ke_1$ with $k\ge 3$ such that $e_1,\cdots ,e_k$ are distinct edges, $\triangle _1,\cdots ,\triangle _k$ are distinct triangles, and $\{e_i,e_{i+1}\}\subseteq \triangle _i$ for each $i\in \{1,2,\cdots ,k\}$, where $e_{k+1}=e_1$. In $\cup _{i=1}^k\triangle _i$, the edges $e_1,e_2,\ldots ,e_k$ are join edges and other edges are non-join edges.

Let $C=e_1\triangle _1e_2\cdots e_k\triangle _ke_1$ be a triangle-cycle. We call C odd if its length k is odd. By abusing notations, we identify C with the graph $\cup _{i=1}^k\triangle _i$, whose edge set we denote as E(C). We write $J_{C}=\{e_1,\cdots ,e_k\}$ for the set of join edges, and $N_{C}=E(C)\backslash J_{C}$ for the set of non-join edges. Let $\mathscr {T}_{C}$ denote the set of triangles in C. A triangle in $\mathscr {T}_{C}$ is basic if it belongs to $\mathscr {B}_{C}=\{\triangle _1,\cdots ,\triangle _k\}$. Two basic triangles $\triangle _i$ and $\triangle _j$ are consecutive if $|i-j|\in \{1,k-1\}$. Triangles in $\mathscr {T}_{C}$ can be classified into four categories:

$$\begin{aligned} \mathscr {T}_{C,i}=\{\triangle \in \mathscr {T}_{C}:|\triangle \cap J_{C}|=i\},\quad i=0,1,2,3. \end{aligned}$$

It is clear from Definition 1 that $\mathscr {B}_{C}\subseteq \mathscr {T}_{C,2}\cup \mathscr {T}_{C,3}$. We will establish a strengthening $\mathfrak B\varsubsetneq \mathfrak M\varsubsetneq \mathfrak I$ of the inclusion relations (1.5). The proof needs the following equivalence implied by hypergraph theory.

Lemma 1

Let G be a graph. Then $G\in \mathfrak B$ if and only if every odd triangle-cycle C in G (if any) contains a basic triangle that belongs to $\mathscr {T}_{C,3}$;

Proof

Recall that $G\in \mathfrak B$ if and only if hypergraph $\mathcal H_G=(E(G),\varLambda (G))$ is balanced. By definition, the balance condition amounts to saying that every odd triangle-cycle C in G (if any) has a triangle $\triangle$ which contains at least 3 joins. It must be the case that $\triangle$ is formed by exactly 3 joins, giving $\triangle \in \mathscr {T}_{C,3}$.    $\square$

Observe that the balanced, Mengerian, and integral properties are all closed under taking subgraphs (see, e.g., Theorems 78.2 and 79.1 of [12]).

Lemma 2

Let G be a graph and H a subgraph of G. If $G\in \mathfrak X$ for some $\mathfrak X\in \{\mathfrak B,\mathfrak M,\mathfrak I\}$, then $H\in \mathfrak X$.   $\square$

Lemma 3

$K_4\in \mathfrak I\setminus \mathfrak M$.

Proof

Note that $K_4\not \in \mathfrak M$ follows from the fact that $\tau (K_4)=2$ and $\nu (K_4)=1$. To see $K_4=(V,E)\in \mathfrak I$, for any $\mathbf x\in \mathbb Q^E$, let $F(\mathbf x)=\{e\in E:0<x(e)<1\}$ consist of “fractional” edges w.r.t $\mathbf x$. Taking arbitrary $\mathbf w\in \mathbb Z_+^{E}$, we consider an optimal fractional triangle cover $\mathbf x^*$ for $(K_4,\mathbf w)$ such that

$$\begin{aligned} F(\mathbf x^*)\,\text {is as small as possible}. \end{aligned}$$

We are done by showing that $\mathbf x^*$ is integral. Suppose it were not the case. The optimality says that $\mathbf w^T\mathbf x^*=\tau ^*_w(K_4)$ and $\mathbf x^*\le \mathbf 1$. Thus $F(\mathbf x^*)\ne \emptyset$.

If $x^*(e)=1$ for some $e\in E$, then $\mathbf x^*|_{E\setminus \{e\}}$ is a fractional triangle cover for $K_4\setminus e$ such that $(\mathbf w|_{E\setminus \{e\}})^T\mathbf x^*|_{E\setminus \{e\}}=\tau _w^*(K_4)-w(e)$. Since $K_4\setminus e\in \mathfrak B\subseteq \mathfrak I$, there is a triangle cover S of $K_4\setminus e$ with minimum weight $w(S)\le \tau _w^*(K_4)-w(e)$. So $S\cup \{e\}$ is a triangle cover of $K_4$ with weight $w(S)+w(e)\le \tau _w^*(K_4)$, and hence the incidence vector $\mathbf x\in \{0,1\}^{E}$ of $S\cup \{e\}$ is an optimal fractional triangle cover for $(K_4,\mathbf w)$ with $F(\mathbf x)=\emptyset \varsubsetneq F(\mathbf x^*)$ contradicting the minimality of $F(\mathbf x^*)$.

Therefore $x^*(e)<1$ for all $e\in E$, and $A_{K_4}\mathbf x^*\ge \mathbf 1$ enforces that every triangle of $K_4$ intersects $F(\mathbf x^*)$ with at least 2 edges. Thus $F(\mathbf x^*)$ contains four edges $e_1,e_2,e_3,e_4$ that induce a cycle of $K_4$, where $\{e_1,e_3\}$ and $\{e_2,e_4\}$ are two matchings of $K_4$. Without loss of generality we may assume that $x^*(e_1)=\min _{i=1}^4x^*(e_i)$. Let $\mathbf x\in \mathbb Q_+^E$ be defined by $x(e_i)=x^*(e_i)+(-1)^ix^*(e_1)$ for $i=1,2,3,4$ and $x(e)=x^*(e)$ for $e\in E\setminus \{e_1,e_2,e_3,e_4\}$. It is straightforward that

$$\begin{aligned} \mathbf w^T\mathbf x=\mathbf w^T\mathbf x^*\,\text {and}\, F(\mathbf x)\subseteq F(\mathbf x^*)\setminus \{e_1\}. \end{aligned}$$

Since every triangle of $K_4$ intersects each of $\{e_1,e_3\}$ and $\{e_2,e_4\}$ with exactly one edge, we have $A_{K_4}\mathbf x=A_{K_4}\mathbf x^*\ge \mathbf 1$, which along with $\mathbf w^T\mathbf x=\mathbf w^T\mathbf x^*$ says that $\mathbf x\in \{0,1\}^{E}$ is an optimal fractional triangle cover for $(K_4,\mathbf w)$. However, $F(\mathbf x) \varsubsetneq F(\mathbf x^*)$ gives a contradiction.    $\square$

Theorem 5

$\mathfrak B\varsubsetneq \mathfrak M\varsubsetneq \mathfrak I$.

Proof

In view of Lemma 3, it suffices to show that the graph $G=(V,E)$ depicted in Fig. 1 belongs to $\mathfrak M\setminus \mathfrak B$. Note that $G=e_1\triangle _1 e_2\cdots e_7\triangle _7e_1$ is an odd triangle-cycle of length 7, where $\mathscr {B}_G=\{\triangle _1,\triangle _2,\ldots ,\triangle _7\}$ and $\varLambda =\varLambda (G)=\mathscr {T}_G=\{\triangle _1,\ldots ,\triangle _7,\triangle _8\}$.

Fig. 1.

Graph $G\in \mathfrak M\setminus \mathfrak B$.

It is routine to check that none of G’s basic triangles $\triangle _1,\triangle _2,\ldots ,\triangle _7$ belongs to $\mathscr {T}_{G,3}$. Hence Lemma 1 asserts that $G\not \in \mathfrak B$. To prove $G\in \mathfrak M$, by Theorem 1, it suffices to prove that, for any $\mathbf w\in \mathbb Z_+^E$ and an optimal solution $\mathbf y^*$ of $\max \{\mathbf 1^T\mathbf y:A_G^{T}\mathbf y\le \mathbf w,\mathbf y\ge \mathbf 0,2\mathbf y \in \mathbb Z_+^{\varLambda }\}$, there is an integral triangle packing $\mathbf z\in \mathbb Z_+^{\varLambda }$ of $(G,\mathbf w)$ such that $\mathbf 1^T\mathbf z\ge \mathbf 1^T\mathbf y^*$.

Let $\mathbf y'\in \{0,1/2\}^{\varLambda }$ be defined by $y'(\triangle )=y^*(\triangle )-\lfloor y^*(\triangle )\rfloor$ for each $\triangle \in \varLambda$, and let $\mathbf w'\in \mathbb Z_+^E$ be defined by $w'(e)=w(e)-\sum _{\triangle \in \varLambda :e\in \triangle }\lfloor y(\triangle )\rfloor$ for each $e\in E$. Then $\mathbf y'$ is a fractional triangle packing of $(G,\mathbf w')$ such that

$$\begin{aligned} \mathbf 1^T\mathbf y'=\mathbf 1^T\mathbf y^*-\sum \nolimits _{\triangle \in \varLambda }y^*(\triangle ). \end{aligned}$$

If there is an integral packing $\mathbf z'$ of $(G,\mathbf w')$ such that $\mathbf 1^T\mathbf z'\ge \mathbf 1^T\mathbf y'$, then $\mathbf z$ with $z(\triangle )=\lfloor y^*(\triangle )\rfloor +z'(\triangle )$ for each $\triangle \in \varLambda$ is an integral packing of $(G,\mathbf w)$ satisfying $\mathbf 1^T\mathbf z\ge \sum _{\triangle \in \varLambda }y^*(\triangle )+\mathbf 1^T\mathbf y'= \mathbf 1^T\mathbf y^*$ as desired. We next show such a $\mathbf z'$ does exist by distinguishing two cases for integral weight $\mathbf w'$.

In case of $w'(e)\ge 1$ for each $e\in E$, we observe that $\mathbf z'$ with $z'(\triangle _i)=1$ for $i=1,3,6,8$ and $z'(\triangle _i)=0$ for $i=2,4,5,7$ is a triangle packing of $(G,\mathbf w')$ with $\mathbf 1^T\mathbf z'=4=|\varLambda |/2\ge \mathbf 1^T\mathbf y'$.

In case of $w'(e)=0$ for some $e\in G$, the restriction $\mathbf y''$ of $\mathbf y'$ to $\varLambda ({G\setminus e})$ is a fractional triangle packing of $(G\setminus e,\mathbf w'|_{E\setminus e})$ with $\mathbf 1^T\mathbf y''=\mathbf 1^T\mathbf y'$. Using Lemma 1, it is routine to check that $G\setminus e\in \mathfrak B$, which along with $\mathfrak B\subseteq \mathfrak M$ gives an integral triangle packing $\mathbf z''$ of $(G\setminus e,\mathbf w'|_{E\setminus e})$ with $\mathbf 1^T\mathbf z''\ge \mathbf 1^T\mathbf y''$. For each triangle $\triangle \in \varLambda$, set $z'(\triangle )$ to 0 if $e\in \triangle$ and to $z''(\triangle )$ otherwise. It follows that $\mathbf z'\in \mathbb Z_+^{\varLambda }$ is an integral triangle packing of $(G,\mathbf w')$ with $\mathbf 1^T\mathbf z'=\mathbf 1^T\mathbf z''\ge \mathbf 1^T\mathbf y'$ as desired.    $\square$

Lemma 4

If C is an odd triangle-cycle of graph $G\in \mathfrak I$, then C contains either a basic triangle belonging to $\mathscr {T}_{C,3}$ or a non-basic triangle belonging to $\mathscr {T}_{C,0}\cup \mathscr {T}_{C,1}$.

Proof

By contradiction, suppose that graph $G\in \mathfrak I$ and its odd triangle-cycle C of length $2k+1$ form a counterexample, i.e., $\mathscr {B}_C\subseteq \mathscr {T}_{C,2}$ and $\mathscr {T}_C\setminus \mathscr {B}_C\subseteq \mathscr {T}_{C,2}\cup \mathscr {T}_{C,3}$. By Observation 2, we have $C\in \mathfrak I$. Let $\mathbf w\in \{1,\infty \}^{E(C)}$ be defined by $w(e)=1$ for all $e\in J_C$ and $w(e)=\infty$ for all $e\in N_C$. On one hand, $\mathscr {B}_C\subseteq \mathscr {T}_{C,2}$ implies that each join edge of C exactly belongs to two basic triangles. To break all $2k+1$ basic triangles, we have to delete at least $k+1$ join edges unless we use some non-join edge (with infinity weight). Thus $\tau _w(C)\ge k+1$.

On the other hand, note that every triangle of C contains at least two join edges in $J_C$. Thus $\mathbf x\in \{1/2,0\}^{E(C)}$ with $x(e)=1/2$ if $e\in J_C$ and $x(e)=0$ otherwise is a fractional triangle cover of C. This along with $|J_C|=2k+1$ and $\mathbf w|_{J_C}=\mathbf 1$ shows that $\tau ^{*}_w(C)\le |J_C|/2 = k+1/2$. However, $\tau _w(C)> \tau ^{*}_w(C)$ contradicts $C\in \mathfrak I$.    $\square$

The concept of triangle graph provides an efficient tool for studying triangle covering. Suppose that G is a graph with at least a triangle. Its triangle graph, denoted as T(G), is a graph whose vertices are named as triangles of G such that $\triangle _i\triangle _j$ is an edge in T(G) if and only if $\triangle _i$ and $\triangle _j$ are distinct triangles in G which share a common edge. For example, the graph G in Fig. 1 has its triangle graph as depicted in Fig. 2.

Fig. 2.

The triangle graph T(G) of G in Fig. 1.

A graph $G\not \in \mathfrak B$ is called minimal if every proper subgraph H of G belongs to $\mathfrak B$. Let $\mathfrak N$ denote the set of these minimal graphs.

Theorem 6

If $G\in \mathfrak N$, then G is either $K_{4}$ or an odd triangle-cycle with length at least 5 such that $\mathscr {B}_G\subseteq \mathscr {T}_{G,2}$ and $\mathscr {T}_{G}\setminus \mathscr {B}_{G}\subseteq \mathscr {T}_{G,1}\cup \mathscr {T}_{G,3}$.

Proof

Clearly, $K_{4}\in \mathfrak N$. So we consider $G\ne K_{4}$. Since $G\not \in \mathfrak B$ is minimal, G is $K_{4}$-free, and by Lemma 1, $G=e_1\triangle _1e_2\cdots e_k\triangle _ke_1$ is an odd triangle-cycle such that $\mathscr {B}_G\subseteq \mathscr {T}_{C,2}$, where $k\ge 5$ is odd. Observe that triangle-cycle G corresponds to a cycle $\tilde{C}=\tilde{e}_1{\triangle _1}\tilde{e}_2\cdots \tilde{e}_k{\triangle _k}\tilde{e}_1$ in triangle graph T(G). We first present a series of useful properties.

Property 1

If ${\triangle _i}{\triangle _j}$ is a chord of $\tilde{C}$, then the common edge of $\triangle _i$ and $\triangle _j$ is an non-join edge.

Since $\{\triangle _i,\triangle _j\}\subseteq \mathscr {T}_{G,2}$ and they are not consecutive in G, $\triangle _i\cap J_G$ and $\triangle _j\cap J_G$ are disjoint.    $\blacksquare$

Property 2

If both $\triangle _i \triangle _j$ and ${\triangle _j} {\triangle _k}$ are chords of $\tilde{C}$, then $\triangle _i,\triangle _j,\triangle _k$ share the same non-join edge in G, and ${\triangle _i} {\triangle _k}$ is a chord of $\tilde{C}$.

It follows from Property 1 that each of $\triangle _i,\triangle _j,\triangle _k$ has only one non-join edge.   $\blacksquare$

Property 3

If $\triangle _{i_1},\triangle _{i_2},\ldots ,\triangle _{i_t}$ are all basic triangles in $\mathscr {B}_G$ that contain $e\in N_{G}$, where $t\ge 2$ and $i_1<i_2<\cdots <i_t$, then for each $j=1,2,\ldots ,t$, $|\{\triangle _{i_j},\triangle _{i_j+1}\ldots ,\triangle _{i_{j+1}-1},\triangle _{i_{j+1}}\}|$ is even (where $i_{t+1}=i_1$ in case of $j=t$).

Otherwise, $C_j=e\triangle _{i_j}e_{i_j+1}\triangle _{i_j+1}\cdots \triangle _{i_{j+1}-1}e_{i_{j+1}}\triangle _{i_{j+1}}e$ is an odd triangle-cycle of G for some $1\le j\le t$. Observe that every basic triangle of $C_j$ belongs to $\mathscr {T}_{C_j,2}$. Thus Lemma 1 says that $C_j\not \in \mathfrak B$, which along with the minimality of $G\in \mathfrak N$ enforces that $C_j=G$. However this is absurd because $C_j$ does not contain the join edge $e_{i_{j+2}}\in J_G$ of G.   $\blacksquare$

Property 4

For each $e\in N_{G}$, there are exactly an odd number of basic triangles in $\mathscr {B}_G$ that contain e.

Since G is the union of its basic triangles, e is contained by some basic triangle of G. The property is instant from Property 3 and the odd length k of the triangle-cycle G.   $\blacksquare$

We now proceed to prove $\mathscr {T}_{G}\setminus \mathscr {B}_{G}\subseteq \mathscr {T}_{G,1}\cup \mathscr {T}_{G,3}$. Suppose for a contradiction that there exists $\triangle \in \mathscr {T}_G\setminus \mathscr {B}_G$ with $\triangle \in \mathscr {T}_{G,0}$. Then $\triangle$ consists of three non-join edges $p,q,r\in N_G$. Let

$$\begin{aligned} \mathscr {B}_p=\{\triangle \in \mathscr {B}_G:p\in \triangle \}, \mathscr {B}_q=\{\triangle \in \mathscr {B}_G:q\in \triangle \}, \mathscr {B}_r=\{\triangle \in \mathscr {B}_G:r\in \triangle \} \end{aligned}$$

denote the sets of basic triangles (of G) that contain p, q, r, respectively. Notice from Property 4 that

$$\begin{aligned} |\mathscr {B}_p|, |\mathscr {B}_q|\,\text {and}\, |\mathscr {B}_r|\,\text {are odd numbers}. \end{aligned}$$

We distinguish between two cases depending on whether all of $\mathscr {B}_p,\mathscr {B}_q,\mathscr {B}_r$ are singletons or not.

Case 1. $|\mathscr {B}_p|=|\mathscr {B}_q|=|\mathscr {B}_r|=1$. We may assume without loss of generality that $\mathscr {B}_j=\{\triangle _{i_j}\}$ for $j\in \{p,q,r\}$ and $i_p< i_q< i_r$. Note that

$$\begin{aligned} C_{pq}= & {} p\triangle _{i_p}e_{i_p+1}\triangle _{i_p+1}\cdots e_{i_q}\triangle _{i_q}q\triangle p,\\ C_{qr}= & {} q\triangle _{i_q}e_{i_q+1}\triangle _{i_q+1}\cdots e_{i_r}\triangle _{i_r}r\triangle q, \\ C_{rp}= & {} r\triangle _{i_r}e_{i_r+1}\triangle _{i_r+1}\cdots e_{i_p}\triangle _{i_p}p\triangle r \end{aligned}$$

are triangle-cycles of G whose basic triangles each contain exactly two join edges. Observe that the sum of lengths of $C_{pq},C_{qr},C_{rp}$ equals $k+6$, which is odd. So at least one of $C_{pq},C_{qr},C_{rp}$, say $C_{pq}$, has an odd length. It follows from $\mathscr {B}_{C_{pq}}\subseteq \mathscr {T}_{C_{pq},2}$ and Lemma 1 that $C_{pq}\not \in \mathfrak B$. Now the minimality of $G\in \mathfrak N$ enforces $C_{pq}=G$. Hence the join edge $e_{i_q+2}\in J_G$ must be one of $e_{i_p},e_{i_p+1},\ldots ,e_{i_q-1}$, from which we deduce that $e_{i_q+2}=e_{i_p}$ (and $i_q+1=i_r$). As $e_{i_q+2}$ has a common vertex with $e_{i_q}$, it follows that $e_{i_p}$, $e_{i_q+1}$ and r form a triangle, and $p,q,r, e_{i_p}, e_{i_p+1}, e_{i_q+1}$ induce a $K_4$, contradicting the fact that G is $K_4$-free.

Case 2. $\max \{|\mathscr {B}_p|,|\mathscr {B}_q|,|\mathscr {B}_r|\}\ge 3$. Suppose without loss of generality that $\mathscr {B}_p=\{\triangle _{i_1},\cdots ,\triangle _{i_t}\}$ where $t\ge 3$ and $i_1<i_2\cdots <i_t$. Setting $i_{t+1}=i_1$, since $\mathscr {B}_p\cap \mathscr {B}_q=\emptyset$, we have $|\mathscr {B}_q|=\sum _{j=1}^t|\{\triangle _{i_j},\triangle _{i_{j}+1}\cdots ,\triangle _{i_{j+1}}\}\cap \mathscr {B}_q|$. Recall that $|\mathscr {B}_q|$ is odd. So there exists $j\in \{1,\ldots ,t\}$ such that $\{\triangle _{i_j},\triangle _{i_{j}+1}\cdots ,\triangle _{i_{j+1}}\}\cap \mathscr {B}_q$ consists of

$$\begin{aligned} \text {an odd number}\,s\,\text {of basic triangles}\, \triangle _{h_1},\ldots ,\triangle _{h_s}, \end{aligned}$$

where $i_j<h_1<\cdots<h_s<i_{j+1}$. By Property 3, $|\{\triangle _{i_j}, \triangle _{i_j+1}\ldots , \triangle _{i_{j+1}}\}|$ is even, and $|\{\triangle _{h_{\ell }},\triangle _{h_{\ell }+1},\ldots ,\triangle _{h_{\ell +1}}\}|$ is even for each $\ell \in \{1,\ldots ,s-1\}$. Note that

$$\begin{aligned}&|\{\triangle _{i_j}, \triangle _{i_j+1}\ldots , \triangle _{i_{j+1}}\}|\\= & {} |\{\triangle _{i_j}, \triangle _{i_j+1}\ldots ,\triangle _{h_1}\}|+\left( \sum _{\ell =1}^{s-1}|\{\triangle _{h_{\ell }},\triangle _{h_{\ell }+1},\ldots ,\triangle _{h_{\ell +1}}\}|\right) \\&+ |\{\triangle _{h_s},\triangle _{h_s+1}\ldots ,\triangle _{i_{j+1}}\}|-s\\\equiv & {} (h_1-i_j)+(i_{j+1}-h_s)-s\pmod 2 \end{aligned}$$

Since s is odd, either $h_1-i_j$ or $i_{j+1}-h_s$ is odd. Suppose by symmetry that $h_1-i_j$ is odd. It follows that $C= p\triangle _{i_j}e_{i_j+1}\triangle _{i_j+1}\cdots e_{h_1}\triangle _{h_1}q\triangle p$ is a triangle-cycle of G such that $\mathscr {B}_C\subseteq \mathscr {T}_{C,2}$. As the length $h_1-i_j+2$ is odd, we deduce from Lemma 1 that $C\not \in \mathfrak B$. In turn $G\in \mathfrak N$ enforces $C=G$. Similar to Case 1, $e_{h_1+2}\in J_G\subseteq C$ implies that $\triangle _{h_1}, \triangle _{i_{j+1}},\triangle$ form a $K_{4}$, a contradiction to the $K_4$-freeness of G. The contradiction shows that $(\mathscr {T}_{G}\setminus \mathscr {B}_{G})\cap \mathscr {T}_{G,0}=\emptyset$.

It remains to prove $(\mathscr {T}_{G}\setminus \mathscr {B}_{G})\cap \mathscr {T}_{G,2}=\emptyset$. Suppose on the contrary that there exists $\triangle \in \mathscr {T}_G\setminus \mathscr {B}_G$ which consists of two join edges $p,q\in J_G$ and one non-join edge $r\in N_G$. Again we set $\mathscr {B}_p=\{\triangle \in \mathscr {B}_G:p\in \triangle \}$, $\mathscr {B}_q=\{\triangle \in \mathscr {B}_G:q\in \triangle \}$ and $\mathscr {B}_r=\{\triangle \in \mathscr {B}_G:r\in \triangle \}$. Recalling $\mathscr {B}_G\subseteq \mathscr {T}_{G,2}$, we derive $|\mathscr {B}_p|=|\mathscr {B}_q|=2$. Suppose without loss of generality that $\mathscr {B}_p=\{\triangle _{i_p},\triangle _{i_p+1}\}$, $\mathscr {B}_q=\{\triangle _{i_q},\triangle _{i_q+1}\}$ and $i_p< i_p+1< i_q< i_q+1$ (note $p=e_{i_p+1}, q=e_{i_q+1}$). Recall from Property 4 that $|\mathscr {B}_r|$ is an odd number. Observe that both $C= p\triangle _{i_p+1}e_{i_p+2}\triangle _{i_p+2}\cdots e_{i_q}\triangle _{i_q}q\triangle p$ and $C'= q\triangle _{i_q+1}e_{i_q+2}\triangle _{i_q+2}\cdots e_{i_p}\triangle _{i_p}p\triangle q$ are triangle-cycles whose basic triangles each contain exactly 2 join edges. Because the length of G is odd, exactly one of C and $C'$, say C, whose length is odd. By Lemma 1(i), $C\not \in \mathfrak B$. In turn $G\in \mathfrak N$ gives $C=G$. Since neither $\triangle _{i_p}$ nor $\triangle _{i_q+1}$ is a basic triangle of C and $\triangle _{i_p}\ne \triangle _{i_q+1}$, we derive that $e_{i_q+2}\in G\setminus C$, a contradiction to $C=G$. This completes the proof of Theorem 6.    $\square$

Let $\mathfrak X\in \{\mathfrak M,\mathfrak I\}$. If graph $G\in \mathfrak X\setminus \mathfrak B$ is minimal in the sense that every proper subgraph H of G is outside $\mathfrak X\setminus \mathfrak B$, then $H\in \mathfrak X$ (by Lemma 2) enforces $H\in \mathfrak B$. Hence $G\in \mathfrak N$. Conversely, if $G\in \mathfrak X\cap \mathfrak N$, then every subgraph H of G satisfies $H\in \mathfrak B\subseteq \mathfrak X$, giving $H\not \in \mathfrak X\setminus \mathfrak B$. Thus the set of minimal graphs in $\mathfrak X\setminus \mathfrak B$ is

$$\begin{aligned} \{G\in \mathfrak X\setminus \mathfrak B: H\not \in \mathfrak X\setminus \mathfrak B\text { for every }H\varsubsetneq G\}=\mathfrak N\cap \mathfrak X, \text { where }\mathfrak X\in \{\mathfrak M,\mathfrak I\}. \end{aligned}$$

(2.1)

Corollary 1

If $G\in \mathfrak N\cap \mathfrak I$ (i.e., $G\in \mathfrak I\setminus \mathfrak B$ is minimal) , then G is either $K_4$ or an odd triangle-cycle such that $\mathscr {B}_G\subseteq \mathscr {T}_{G,2}$, $\mathscr {T}_{G}\setminus \mathscr {B}_{G}\subseteq \mathscr {T}_{G,1}\cup \mathscr {T}_{G,3}$, and $\mathscr {T}_{G,1}= \mathscr {T}_{G,1}\setminus \mathscr {B}_G\ne \emptyset$.

Proof

In view of Theorem 6, it suffices to consider G being an odd triangle-cycle such that $\mathscr {B}_G\subseteq \mathscr {T}_{G,2}$ and $\mathscr {T}_{G}\setminus \mathscr {B}_{G}\subseteq \mathscr {T}_{G,1}\cup \mathscr {T}_{G,3}$. In turn, Lemma 4 implies the existence of at least a non-basic triangle of G that belongs to $\mathscr {T}_{G,1}$.    $\square$

Corollary 2

If $G\in \mathfrak N\cap \mathfrak M$ (i.e., $G\subseteq \mathfrak M\setminus \mathfrak B$ is minimal) , then G is an odd triangle-cycle such that $\mathscr {B}_G\subseteq \mathscr {T}_{G,2}$, $\mathscr {T}_{G}\setminus \mathscr {B}_{G}\subseteq \mathscr {T}_{G,1}\cup \mathscr {T}_{G,3}$, and $\mathscr {T}_{G,1}= \mathscr {T}_{G,1}\setminus \mathscr {B}_G\ne \emptyset$.

Proof

Note from $G\in \mathfrak M$ that $G\ne K_4$. As $\mathfrak M\subseteq \mathfrak I$, the conclusion is immediate from Corollary 1.    $\square$

3 Planar Graphs

In this section, we study the planar case more closely, and characterize planar graphs in $\mathfrak M$ by excluding pseudo-wheels defined as follows.

Definition 2

A triangle-cycle C is a pseudo-wheel if it has length at least 4, $\mathscr {T}_{C}=\mathscr {B}_{C}$ and each pair of non-consecutive basic triangles of C is edge-disjoint.

It is easy to see that a triangle-cycle C is a pseudo-wheel if and only if its triangle graph T(C) is an induced cycle with length at least 4. Thus every wheel other than $K_4$ is a pseudo-wheel. Two pseudo-wheels that are not wheels are shown in Fig. 3.

Fig. 3.

Examples of pseudo-wheels.

Lemma 5

If C is an odd pseudo-wheel, then $C\not \in \mathfrak I$.

Proof

Suppose the length of C is $2k+1$. Let $\mathbf w\in \mathbb Z_+^E(C)$ be defined by $w(e)=1$ for all $e\in J_C$ and $w(e)=\infty$ for all $e\in N_C$. Then $\tau _w(C)=k+1$. On the other hand $\mathbf x\in \{0,1/2\}^{E(C)}$ with $x(e)=1/2$ for all $e\in J_C$ and $x(e)=0$ for all $e\in N_C$ is a fractional triangle cover of C, showing $\tau _w^*(C)\le \mathbf w^T\mathbf x=k+1/2<\tau _w(C)$.    $\square$

If $\triangle ^i,\triangle ^o,\triangle$ are distinct triangles of plane graph G such that $\triangle ^i$ is inside $\triangle$ and $\triangle ^o$ is outside $\triangle$, then we say that $\triangle$ is a separating triangle of $\triangle ^i$ and $\triangle ^o$, or $\triangle$ separates $\triangle ^i$ from $\triangle ^o$.

A triangle-path in graph G is a sequence $P=\triangle _1e_1\cdots e_k\triangle _{k+1}$ with $k\ge 1$ such that $e_1,\cdots ,e_k$ are distinct edges, $\triangle _1,\cdots ,\triangle _{k+1}$ are distinct triangles of G, and $\{e_1\}\subseteq \triangle _1, \{e_k\}\subseteq \triangle _{k+1}, \{e_i,e_{i+1}\}\subseteq \triangle _{i+1}$ for each $i\in [k-1]$. In $\cup _{i=1}^{k+1}\triangle _i$, the edges $e_1,e_2,\ldots ,e_k$ are called join edges and other edges are called non-join edges. Let $J_P$ denote the set of join edges of P. The length of P is defined as k. We often say that P is a triangle-path from $\triangle _1$ to $\triangle _{k+1}$.

Lemma 6

Let G be a plane graph in which $\triangle$ is a separating triangle of triangles $\triangle ^i$ and $\triangle ^o$. Then $\triangle$ contains at least one join edge of every triangle-path from $\triangle ^i$ to $\triangle ^o$ in G.

Proof

Consider an arbitrary triangle-path $P=\triangle _1e_1\cdots e_k\triangle _{k+1}$ in G from $\triangle _1=\triangle ^i$ to $\triangle _{k+1}=\triangle ^o$. We prove $\triangle \cap \{e_1,\ldots ,e_k\}\ne \emptyset$ by induction on k. The basic case of $k=1$ is trivial. We consider $k\ge 2$ and assuming that the lemma holds when triangle-path involved has length at most $k-1$. If $\triangle _2=\triangle$, then we are done. If $\triangle _2\ne \triangle$, then either $\triangle$ separates $\triangle _1$ from $\triangle _2$ or separates $\triangle _2$ and $\triangle _{k+1}$. Observe that $\triangle _1e_1\triangle _2$ is a triangle-path of length $1<k$, and $\triangle _2e_2\ldots ,e_{k}\triangle _{k+1}$ is a triangle-path of length $k-1$. From the induction hypothesis, we derive $e_1\in \triangle$ in the former case, and $e_j\in \triangle$ for some $j=2,\ldots ,k$ in the latter case.    $\square$

Lemma 7

Let $C=e_1\triangle _1e_2\cdots e_k\triangle _ke_1$ with $k\ge 3$ be a triangle-cycle. If C is plane and $\mathscr {B}_{C}\subseteq \mathscr {T}_{C,2}$, then $\triangle _h$ does not separate $\triangle _i$ from $\triangle _j$ for any distinct $h,i,j\in \{1,\ldots ,k\}$.

Proof

Note that C contains a triangle-path P from $\triangle _i$ and $\triangle _j$ with $J_P\subseteq J_C\setminus \triangle _h$. The triangle-path P along with Lemma 6 implies the result.    $\square$

Theorem 7

If C is a planar triangle-cycle such that $\mathscr {B}_{C}\subseteq \mathscr {T}_{C,2}$, then $\mathscr {T}_{C}\subseteq \mathscr {T}_{C,0}\cup \mathscr {T}_{C,2}$.

Proof

Suppose that $C=e_1\triangle _1e_2\cdots e_k\triangle _ke_1$ with $k\ge 3$ is plane, and there exists $\triangle \in \mathscr {T}_C$ with $\triangle \in \mathscr {T}_{C,1}\cup \mathscr {T}_{C,3}$.

Case 1. $\triangle \in \mathscr {T}_{C,3}$ consists of three join edges $e_h,e_i,e_j$, where $1\le h<i<j\le k$. The structure of the triangle graph T(C) is illustrated in the left part of Fig. 4.

Fig. 4.

The triangle graph T(C) in the two cases of the proof for Theorem 7.

Fig. 5.

$\triangle _{h-1}$ and $\triangle _h$ are both inside $\triangle$.

For each pair $(s,t)\in \{(h,i-1),(i,j-1),(j,h-1)\}$, there is a triangle-path in C from $\triangle _s$ to $\triangle _t$ whose set of join edges is disjoint from $\{e_h,e_i,e_j\}=\triangle$. It follows from Lemma 6 that

$$\begin{aligned} \begin{array}{ll} \triangle \, \text {does not separate}\, \triangle _{s}\, \text {from}\, \triangle _{t}\, \text {for each }\\ (s,t)\in \{(h,i-1),(i,j-1),(j,h-1)\}. \end{array} \end{aligned}$$

(3.1)

Suppose that $\triangle$ separates $\triangle _{h-1}$ from $\triangle _h$, and separates $\triangle _{i-1}$ from $\triangle _i$. Without loss of generality let $\triangle _{h-1}$ and $\triangle _h$ sit inside and outside $\triangle$, respectively. Then (3.1) implies that $\triangle _j$ and $\triangle _{i-1}$ are inside and outside $\triangle$, respectively. In turn, i is inside $\triangle$, and (3.1) says that $\triangle _{j-1}$ is inside $\triangle$. Now $\triangle _{j-1}$ and $\triangle _j$ are both inside $\triangle$, i.e., $\triangle$ does not separate $\triangle _{j-1}$ from $\triangle _j$. Hence, by symmetry we may assume that $\triangle$ does not separate $\triangle _{h-1}$ from $\triangle _h$, and further that $\triangle _{h-1}$ and $\triangle _h$ are both inside $\triangle$. as illustrated in Fig. 5.

As $e_h\in \triangle _{h-1}\cap \triangle _h$, it is easy to see that either $\triangle _{h-1}$ separates $\triangle _h$ from $\triangle _i$ or $\triangle _h$ separates $\triangle _{h-1}$ from $\triangle _i$. The contradiction to Lemma 7 finishes our discussion on Case 1.

Case 2. $\triangle \in \mathscr {T}_{C,1}$ consist of join edge $e_h$ of C (shared with $\triangle _{h-1},\triangle _h$), non-join edge f (shared with $\triangle _i$) and non-join edge g (shared with $\triangle _j$), where h, i, j are distinct. See the right part of Fig. 4. Similar to Case 1, it can be derived from Lemma 6 that

$$\begin{aligned} \triangle \,\text {does not separate} \triangle _{s}\, \text {from}\, \triangle _{t}\, {\text {for each}} (s,t)\in \{(h,i),(i,j),(j,h-1)\}. \end{aligned}$$

Therefore $\triangle$ does not separate $\triangle _{h-1}$ and $\triangle _{h}$. Suppose without loss of generality that both $\triangle _{h-1}$ and $\triangle _{h}$ are inside $\triangle$. Then C has one of the structures as illustrated in Fig. 5 with f in place of $e_i$ and g in place of $e_j$. Again, either $\triangle _{h-1}$ separating $\triangle _h$ from $\triangle _i$ or $\triangle _h$ separating $\triangle _{h-1}$ from $\triangle _i$ contradicts to Lemma 7. This completes the proof.    $\square$

Theorem 8

Let G be a planar graph. Then $G\in \mathfrak N$ if and only if G is $K_{4}$ or an odd pseudo-wheel.

Proof

Sufficiency: Clearly $K_{4}\in \mathfrak N$. If G is an odd pseudo-wheel C, then G is an odd triangle-cycle such that $\mathscr {B}_G\subseteq \mathscr {T}_{G,2}$. By Lemma 1, $G\not \in \mathfrak B$. Since the triangle graph T(C) is an induced cycle, every proper subgraph of C is triangle-cycle-free, and hence belongs to $\mathfrak B$, giving $G\in \mathfrak N$.

Necessity: Suppose that $G\in \mathfrak N$ and $G\ne K_{4}$. By Theorem 6, an odd triangle-cycle with length at least 5 such that $\mathscr {B}_G\subseteq \mathscr {T}_{G,2}$ and $\mathscr {T}_{G}\setminus \mathscr {B}_{G}\subseteq \mathscr {T}_{G,1}\cup \mathscr {T}_{G,3}$. In turn, Theorem 7 enforces

$$\begin{aligned} \mathscr {T}_{C}=\mathscr {B}_{C}. \end{aligned}$$

Suppose for a contradiction that there exists non-consecutive triangles $\triangle _i,\triangle _j\in \mathscr {B}_G$ that share a common non-join edge e, where $i<j-1$. Then G contains two triangle-cycles $C_1=e\triangle _ie_{i+1}\triangle _{i+1}\cdots e_{j}\triangle _je$ and $C_2=e\triangle _je_{j+1}\triangle _{j+1}\cdots e_{i}\triangle _ie$. Because G is odd, one of $C_1$ and $C_2$, say $C_1$, is odd. As $C_1$ is a proper subgraph of $G\in \mathfrak N$, we have $C_1\in \mathfrak B$. By Lemma 1, there exists a basic triangle $\triangle _{h}$ in $\mathscr {B}_{C_{1}}\cap \mathscr {T}_{C_{1},3}$. Because $\triangle _{h}\in \mathscr {T}_{G,2}$, it must be the case that $e\in \triangle _h$. Thus $\triangle _{i},\triangle _{j},\triangle _h$ share a common non-join edge e of G. However in any planar embedding for G, there is one triangle in $\{\triangle _{i},\triangle _{j},\triangle _h\}$, which is a separating triangle of the other two. This is a contradiction to Lemma 7. Thus each pair of non-consecutive basic triangles of G is edge-disjoint, and G is an odd pseudo-wheel.    $\square$

Theorem 9

Let G be a planar graph, then the following are equivalent:

• (i) $G\in \mathfrak B$;

• (ii) $G\in \mathfrak M$;

• (iii) $G\in \mathfrak I$ is $K_4$-free; and

• (iv) G is $K_4$-free and odd pseudo-wheel free.

Proof

Recalling (1.5) and Lemma 3, $\mathfrak B\subseteq \mathfrak M\subseteq \mathfrak I$ and $K_4\in \mathfrak I\setminus \mathfrak M$ imply the relation $(i)\Rightarrow (ii)\Rightarrow (iii)$. If G contains an odd pseudo-wheel H, then $H\not \in \mathfrak I$ by Lemma 5, which along with Lemma 2 would give $G\not \in \mathfrak I$. So we have $(iii)\Rightarrow (iv)$.

It remains to prove $(iv)\Rightarrow (i)$. If $G\not \in \mathfrak {B}$, we take $H\subseteq G$ to be minimal, i.e., $H\in \mathfrak N$. Theorem 8 says that H is $K_{4}$ or an odd pseudo-wheel, i.e., G is not $K_4$-free and G is not odd pseudo-wheel free.    $\square$

4 Remarks

Lemma 4 provides us a necessary condition for $G\in \mathfrak I$ as follows:

$$\begin{aligned} (\mathscr {B}_C\cap \mathscr {T}_{C,3})\cup ((\mathscr {T}_{C,0}\cup \mathscr {T}_{C,1})\setminus \mathscr {B}_C)\ne \emptyset \text { for any odd triangle-cycle }C\text { of }G. \end{aligned}$$

(4.1)

It would be interesting to see if the condition is sufficient for $G\in \mathfrak I$. A supporting evidence is the following.

Remark 1

Condition (4.1) is a necessary and sufficient condition for $K_4$-free planar graph G to be a member of $\mathfrak I$.

Proof

By Theorem 9, a $K_{4}$-free planar graph $G\in \mathfrak I$ implies $G\in \mathfrak B$, and thus $\mathscr {B}_C\cap \mathscr {T}_{C,3}\ne \emptyset$ for every odd triangle-cycle C in G. On the other hand, given a $K_{4}$-free planar graph G satisfying (4.1), we see from Definition 2 that G does not contain any odd pseudo-wheel. It follows from Theorem 8 that G does not contain any subgraph in $\mathfrak N$, which implies $G\in \mathfrak B\subseteq \mathfrak I$.    $\square$

As $\mathfrak M\subseteq \mathfrak I$, condition (4.1) is also necessary for $G\in \mathfrak M$, but it is not sufficient for the total dual integrality. This can be seen from $K_4\not \in \mathfrak M$, which satisfies (4.1): $K_4$ has four odd triangle-cycles with length 3 each containing a triangle without any join edge, and for each odd triangle-cycle C, there is a triangle in $\mathscr {T}_C\setminus \mathscr {B}_C$ that belongs to $\mathscr {T}_{C,0}$. This motivates us to ask about the necessity and sufficiency of the following conditions for $G\in \mathfrak M$:

$$\begin{aligned} (\mathscr {B}_C\cap \mathscr {T}_{C,3})\cup ( \mathscr {T}_{C,1} \setminus \mathscr {B}_C)\ne \emptyset \text { for any odd triangle-cycle }C\text { of }G. \end{aligned}$$

(4.2)

Note that condition (4.2) implies G contains neither $K_4$ nor odd pseudo-wheels. Similar to Remark 1, Theorems 8 and 9 provide the following fact.

Remark 2

Condition (4.2) is a necessary and sufficient condition for planar graph G to be a member of $\mathfrak M$.   $\square$

Appendix: A List of Mathematical Symbols

$(G,\mathbf w)$	Weighted graph $G=(V(G),E(G))$ with $\mathbf w\in \mathbb Z^{E(G)}_+$
$\tau _w(G)$	The minimum weight of an integral triangle cover in $(G,\mathbf w)$
$\nu _w(G)$	The maximum size of an integral triangle packing in $(G,\mathbf w)$
$\tau ^*_w(G)$	The minimum weight of a fractional triangle cover in $(G,\mathbf w)$
$\nu ^*_w(G)$	The maximum size of a fractional triangle packing in $(G,\mathbf w)$
$\tau (G)$	$\tau _w(G)$ when $\mathbf w=\mathbf 1$
$\nu (G)$	$\nu _w(G)$ when $\mathbf w=\mathbf 1$
$A_G$	The triangle-edge incidence matrix of graph G
$\varLambda (G)$	The set of triangles in graph G
$\mathfrak B$	The set of graphs G such that $A_G$ are TUM
$\mathfrak M$	The set of graphs G such that systems $A_Gx\ge 1, x\ge 0$ are TDI
$\mathfrak J$	The set of graphs G such that $\{\mathbf x: A_G\mathbf x\ge 1, \mathbf x\ge \mathbf 0\}$ are intergal
$\mathfrak N$	The set of minimal graphs not belonging to $\mathfrak B$
$\mathscr {T}_C$	The set of triangles in triangle-cycle $C\!=\!e_1\triangle _1e_2\!\cdots \!e_k\triangle _k e_1\!=\!\cup _{i\!=\!1}^k\triangle _i$
$\mathscr {B}_C$	The set of basic triangles in triangle-cycle C, i.e., $\{\triangle _1, \cdots ,\triangle _k\}$
$J_C$	The set of join edges in triangle-cycle C, i.e., $\{e_1,\cdots ,e_k\}$
$N_C$	The set of nonjoin edges in triangle-cycle C, i.e., $E(C)\backslash J_C$
$\mathscr {T}_{C,i}$	$\{\triangle \in \mathscr {T}_C: |\triangle \cap J_C|=i \}$, $i=0,1,2,3$