On the total outer-connected domination in graphs

Abstract

A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph induced by V∖S is connected. The total outer-connected domination number γ _toc (G) is the minimum size of such a set. We give some properties and bounds for γ _toc in general graphs and in trees. For graphs of order n, diameter 2 and minimum degree at least 3, we show that $\gamma_{toc}(G)\le \frac{2n-2}{3}$ and we determine the extremal graphs.

1 Introduction

For domination problems, multiple edges and loops are irrelevant, so we forbid them. We use V(G) and E(G) (or simply V and E) for the vertex set and edge set of a graph G and denote |V(G)|=n, |E(G)|=m. For a vertex v∈V(G), the open neighborhood N(v) is the set {u∈V(G): uv∈E(G)} and the closed neighborhood N[v] is the set N(v)∪{v}. The open neighborhood N(S) of a set S⊆V is the set ⋃ _v∈S N(v), and the closed neighborhood N[S] of S is the set N(S)∪S. For a vertex x∈V∖S, we write N _S (x) for N(x)∩S. A vertex x of V∖S is a private neighbor of a vertex u of S with respect to S, or a (u,S)-private neighbor, if N _S (x)={u}. The degree of a vertex v and the minimum degree in G are respectively denoted by d _G (v) and δ(G) (d(v) and δ for short). We denote by P _n and C _n the path and the cycle on n vertices. We say that we attach a vertex u to a vertex v of a graph if we join u and v by an edge. The 2-corona G of a graph H is obtained by attaching a leaf of a path P ₂ at each vertex of H (all the P ₂’s are distinct and disjoint). The set V(H) is called the handle of G and each attached P ₂ is called a pendant P ₂.

For a graph G, a set S⊆V(G) is a dominating set if N[S]=V(G). A dominating set S is an outer-connected dominating set if the subgraph induced by V∖S, denoted G[V∖S], is connected. The set S⊆V(G) is a total dominating set if N(S)=V(G). A total dominating set S is a total outer-connected dominating set (TOCD set for short) if the subgraph induced by V∖S is connected. The minimum size of a dominating set, a total dominating set, an outer-connected dominating set, a total outer-connected dominating set are the domination number γ(G), the total domination number γ _t (G), the outer-connected domination number γ _oc (G), and the total outer-connected domination number γ _toc (G), respectively. A γ _toc (G)-set (γ _t (G)-set) is a total outer-connected dominating set of G of size γ _toc (G) (a total dominating set of size γ _t (G)). The total outer-connected domination number was introduced by Cyman (2010) and has been studied particularly for trees (see Cyman and Raczek 2009; Hattingh and Joubert 2010).

We note the following results related to the total domination.

If a graph G has k components G ₁,G ₂,…,G _k , then $\gamma_{t}(G) =\sum_{i=1}^{k} \gamma_{t}(G_{i})$. If the connected graph G is not a star, then it admits a γ _t (G)-set containing no leaf of G.

Theorem A

Let H be a connected graph of order n≥3. Then

• 1. (Cockayne et al. 1980) $\gamma_{t}(H)\le\lfloor \frac{2n}{3}\rfloor$.

• 2. (Brigham et al. 2000) $\gamma_{t}(H)= \frac{2n}{3}$ if and only if H is a 2-corona or a cycle C ₃ or C ₆.

  $\gamma_{t}(H)= \frac{2n-1}{3}$ if and only if H is a 2-corona minus a pendant edge or a cycle C ₅.

2 General properties of γ _toc (G)

We begin with some easy properties of γ _toc in graphs of minimum degree at least 1. First it is immediate to check that for paths P _n , cycles C _n , complete graphs K _n , complete bipartite graphs K _p,q with p,q≥2, stars K _1,n−1, we have γ _toc (P _n )=n−1 for 3≤n≤5, γ _toc (P _n )=n−2 for n≥6, γ _toc (C _n )=n−1 for n=3, γ _toc (C _n )=n−2 for n≥4, γ _toc (K _n )=γ _toc (K _p,q )=2, γ _toc (K _1,n−1)=n−1.

From the definition of γ, γ _t , γ _oc and γ _toc , it is clear that for every graph G,

$$1\le\gamma(G)\le \gamma_t(G) \le \gamma_{toc}(G) \le n \quad\mathrm{and} \quad \gamma(G)\le \gamma_{oc}(G) \le \gamma_{toc}(G). $$

It is known that $\frac{\gamma_{t}(G)}{\gamma(G)}\le2$ for every graph. However the other ratios $\frac{ \gamma_{oc}}{\gamma}$, $\frac{ \gamma_{toc}}{\gamma_{t}}$, $\frac{\gamma_{toc}}{\gamma_{oc}}$ are not bounded. For the first two ratios, the star K _1,n−1 satisfies γ=1, γ _t =2, γ _oc =γ _toc =n−1. For the third ratio, we consider for instance an arbitrarily large integer q and a graph G defined from a path $P_{3q}=x_{1}^{1}x_{2}^{1}\cdots x_{q}^{1}x_{1}^{2}x_{2}^{2}\cdots x_{q}^{2} x_{1}^{3}x_{2}^{3}\cdots x_{q}^{3}$ and three vertices y ₁,y ₂,y ₃ by adding the edges $y_{i}x_{j}^{i}$ with i∈{1,3} and 1≤j≤q or i=2 and 1≤j≤4q. For this graph, γ=γ _oc =3, γ _t =6 and γ _toc =q+5.

Proposition 1

Let D be a γ _toc -set of a connected graph G of order n≥3 and minimum degree δ=1. Then D contains all the support vertices of G. If moreover γ _toc (G)≤n−2, then D also contains all the leaves.

Proof

Let v be a support vertex and vu a pendant edge of G. If v∉D, then u∉N[D] or u is an isolated vertex of D, a contradiction. If u∉D, then u is an isolated vertex in G[V∖D], a contradiction except if γ _toc (G)=n−1. □

Proposition 2

Let G ₁,G ₂,…,G _p be the components of a graph G. Then

$$\gamma_{toc}(G)=\min_{1\le i\le p}\biggl\{\sum_{j\neq i}n(G_j)+ \gamma_{toc}(G_i)\biggr\}. $$

Proof

For every TOCD set D of G, V∖D is entirely contained in one component G _i and thus D contains all the components different from G _i . □

Proposition 3

Let G be a graph of order n and minimum degree δ≥1. Then γ _toc (G)=n if and only if each component of G is a K ₂.

Proof

Suppose γ _toc (G)=n and let v be a vertex of degree Δ. The set V∖{v} is an OCD set of G but not a TOCD set since γ _toc (G)=n. Therefore at least one neighbor v′ of v has degree 1 in G. If v has another neighbor in G then V∖{v′} is a TOCD set of G, a contradiction to γ _toc (G)=n. Hence Δ=1 and G is the disjoint union of K ₂’s. Conversely, if each component of G is a K ₂, then γ _toc (G)=n by Proposition 2. □

We already saw that the connected graphs of order n=3, namely K ₃ and P ₃, satisfy γ _toc (G)=n−1. We consider now those of order at least 4.

Proposition 4

Let G be a connected graph of order n≥4. Then γ _toc (G)=n−1 if and only if every edge is pendant or is adjacent to a pendant edge.

Proof

Suppose γ _toc (G)=n−1 and let uv be a non-pendant edge of G with d(v)≥d(u). The set D=V(G)∖{u,v} is an OCD set of G, but not a TOCD set since γ _toc (G)=n−1. Hence some vertex of (N(u)∪N(v))∖{u,v} is isolated in G−{u,v}. If no vertex of N(u)∖N[v] and N(v)∖N[u] is isolated in G−{u,v}, then N(u)∩N(v) contains a vertex z isolated in G−{u,v}. Since n≥4 and d(v)≥d(u), the set V(G)∖{u,z} is a TOCD set of G, in contradiction with γ _toc (G)=n−1. Therefore u or v is the end-vertex of a pendant edge.

Conversely, suppose that every edge of G is pendant or adjacent to a pendant edge. Since n≥4, γ _toc (G)<n by Proposition 3. If every vertex of G is a leaf or a support vertex, then γ _toc (G)=n−1 by Proposition 1. Otherwise, let D be a γ _toc (G)-set, L the set of leaves of G and X the set of the vertices of G which are neither leaves nor support vertices. By the definition of X and G, the neighbors of the vertices of X are support vertices. Hence the set X∪L is independent. Since V∖D⊆X∪L by Proposition 1 and G[V∖D] is connected, |V∖D|=1 and γ _toc (G)=n−1. □

The following corollary is a consequence of Propositions 2, 3 and 4.

Corollary 5

Let G be a graph of order n and minimum degree δ≥1. Then γ _toc (G)=n−1 if and only if each component of G is K ₂, P ₃, C ₃ or is such that every edge is pendant or adjacent to a pendant edge, and at least one component is not K ₂. □

Cyman (2007) gave lower bounds on the outer-connected domination number respectively in terms of n and m and in terms of n and Δ. In Theorems 8 and 13, we establish similar bounds on the total outer-connected domination number.

Definition 6

For 1≤i≤5, let $\mathcal{A}_{i}$ be the family of graphs G=(V,E) defined as follows. The graph G is connected and V is a disjoint union D∪X. Moreover

$\mathcal{A}_{1}$: The components of G[D] are paths P ₂, G[X] is a tree, and every vertex of X has exactly one neighbor in D.

$\mathcal{A}_{2}$: The components of G[D] are paths P ₂ except one which is a path P ₃, G[X] is a tree, and every vertex of X has exactly one neighbor in D.

$\mathcal{A}_{3}$: The components of G[D] are paths P ₂ except one of them which is a path P ₄ or two of them which are paths P ₃, G[X] is a tree, and every vertex of X has exactly one neighbor in D.

$\mathcal{A}_{4}$: The components of G[D] are paths P ₂, G[X] is a tree, and every vertex of X has exactly one neighbor in D except one of them which has two neighbors in D.

$\mathcal{A}_{5}$: The components of G[D] are paths P ₂, G[X] is connected and unicyclic, and every vertex of X has exactly one neighbor in D.

Let $\mathcal{A}=\bigcup_{i=1}^{5}\mathcal{A}_{i}$.

Proposition 7

If $G\in \mathcal{A}$, then $\gamma_{toc}(G)\le \lceil\frac {2(2n-m-1)}{3}\rceil$.

Proof

In each graph G of all families $\mathcal{A}_{i}$, the set D is a TOCD set of G. Hence γ _toc (G)≤|D|. Since |D| is an integer, it is sufficient in order to prove the result to show that |D| is equal to $\frac{4n-2m-2}{3}$, $\frac{4n-2m-1}{3}$, or $\frac{4n-2m}{3}$. In all of these graphs, n=|D|+|X| and m=m(D)+m(D,X)+m(X).

In $\mathcal{A}_{1}$, $m(D)= \frac{|D|}{2}$, m(D,X)=|X| and m(X)=|X|−1. Hence $|D|=\frac{4n-2m-2}{3}$.

In $\mathcal{A}_{2}$, $m(D)= \frac{|D|+1}{2}$, m(D,X)=|X| and m(X)=|X|−1. Hence $|D|=\frac{4n-2m-1}{3}$.

In $\mathcal{A}_{3}$, $m(D)= \frac{|D|}{2}+1$, m(D,X)=|X| and m(X)=|X|−1. Hence $|D|=\frac{4n-2m}{3}$.

In $\mathcal{A}_{4}$, $m(D)= \frac{|D|}{2}$, m(D,X)=|X|+1 and m(X)=|X|−1. Hence $|D|=\frac{4n-2m}{3}$.

In $\mathcal{A}_{5}$, $m(D)= \frac{|D|}{2}$, m(D,X)=|X| and m(X)=|X|. Hence $|D|=\frac{4n-2m}{3}$. □

Theorem 8

Let G be a connected graph with n≥2 vertices and m edges. Then

$$\gamma_{toc}(G) \ge \biggl\lceil\frac{2(2n-m-1)}{3}\biggr\rceil $$

with equality if and only if $G\in \mathcal{A}$.

Proof

Let D be a γ _toc (G)-set and X=V∖D. Since D is a total dominating set and G[X] is connected, we have

(1)

(2)

(3)

Hence $m(G)\ge\frac{|D|}{2}+2|X|-1$ with |X|=n−|D| and thus $m\ge 2n-\frac{3|D|}{2}-1$. Since |D|=γ _toc (G) is an integer, we get $\gamma_{toc}(G) \ge\lceil \frac{2(2n-m-1)}{3}\rceil$.

Assume $\gamma_{toc}(G) = \lceil\frac{2(2n-m-1)}{3}\rceil$. If $\gamma_{toc}(G) = \frac{4n-2m-2}{3}$ then $m(G)= \frac{|D|}{2}+2|X|-1$ (hence |D| is even) and equality holds in (1), (2) and (3). From equality in (1) and since δ(D)≥1, all the components of G(D) are paths P ₂. From equality in (2) and since D is dominating, every vertex of X has exactly one neighbor in D. From equality in (3) and since G[X] is connected, G[X] is a tree. Therefore $G\in \mathcal{A}_{1}$.

If $\gamma_{toc}(G) = \frac{4n-2m-1}{3}$ then $m(G)= \frac{|D|+1}{2}+2|X|-1$ (hence |D| is odd). The unique possibility is that $m(D)= \frac{|D|+1}{2}$ and equality holds in (2) and (3). From $m(D)= \frac {|D|+1}{2}$, all the components of G[D] are paths P ₂ except one which is a path P ₃, from equality in (2) every vertex of X has exactly one neighbor in D and from equality in (3) G[X] is a tree. Therefore $G\in \mathcal{A}_{2}$.

If $\gamma_{toc}(G) = \frac{4n-2m}{3}$ then $m(G)= \frac{|D|}{2}+2|X|$ (hence |D| is even) and we have the following three possibilities. Equality holds in (2) and (3) and $m(D)=\frac{|D|}{2}+1$, in which case $G\in \mathcal{A}_{3}$. Equality holds in (1) and (3) and m(D,X)=|X|+1, in which case $G\in \mathcal{A}_{4}$. Equality holds in (1) and (2) and m(X)=|X|, in which case $G\in \mathcal{A}_{5}$.

Therefore $\gamma_{toc}(G) = \lceil\frac{2(2n-m-1)}{3}\rceil$ implies $G\in \mathcal{A}$. Conversely $G\in \mathcal{A}$ implies $\gamma_{toc}(G) = \lceil\frac{2(2n-m-1)}{3}\rceil$ by Proposition 7, which completes the proof. □

When G is a tree, then m=n−1 and the lower bound of Theorem 8 becomes $\lceil\frac{2n}{3}\rceil$. This bound was already obtained in Cyman (2010). Moreover two different constructions of the extremal trees by recursively applying several operations are given in Cyman (2010) and Hattingh and Joubert (2010). In Corollary 10 we give an explicit description of these trees.

Definition 9

For 1≤i≤4, let $\mathcal{B}_{i}$ be the family of trees T defined as follows and let $\mathcal{B}=\bigcup_{i=1}^{4}\mathcal{B}_{i}$.

$\mathcal{B}_{1}$:  T is the 2-corona of a tree.

$\mathcal{B}_{2}$:  T is obtained from the 2-corona of a tree by replacing a pendant P ₂ by a pendant P ₃.

$\mathcal{B}_{3}$:  T is obtained from the 2-corona of a tree by replacing a pendant P ₂ by a pendant P ₄ or two pendant P ₂ by two disjoint pendant P ₃.

$\mathcal{B}_{4}$:  T is obtained from the 2-corona of a tree by attaching a second pendant P ₂ at one vertex of the handle of the 2-corona.

Corollary 10

Let T be a tree with n≥2 vertices. Then $\gamma_{toc}(T)\ge\lceil\frac {2n}{3}\rceil$ with equality if and only if $T\in \mathcal{B}$.

Proof

By Theorem 8, $\gamma_{toc}(T) \ge\lceil\frac {4n-2m-2}{3}\rceil=\lceil\frac{2n}{3}\rceil$ with equality if and only if T is a tree of $\mathcal{A}$. There is no tree in $\mathcal{A}_{5}$ and the graphs of $\cup_{i=1}^{4}\mathcal{A}_{i}$ are trees if and only if exactly one vertex of each component of D has a neighbor in X (for the components P ₃ and P ₄, this vertex can be a leaf or a support vertex). Therefore for 1≤i≤4, $\mathcal{B}_{i}$ is the class of the trees of $\mathcal{A}_{i}$ and $\mathcal{B}$ is the class of the trees of $\mathcal{A}$. Note that $\gamma_{toc}(T)=\frac{2n}{3}$ if $T\in \mathcal{B}_{1}$, $\gamma_{toc}(T)=\frac{2n+1}{3}$ if $T\in \mathcal{B}_{2}$ and $\gamma_{toc}(T)=\frac{2n+2}{3}$ if $T\in \mathcal{B}_{3} \cup \mathcal{B}_{4}$. □

By Theorem A and Corollary 10 we get the following corollary.

Corollary 11

A tree T satisfies γ _t (T)=γ _toc (T) if and only if it is the 2-corona of a tree.

Our second lower bound on γ _toc is an easy lower bound on γ _t which remains sharp for γ _toc .

Definition 12

Let k≥1 be an integer and let $\mathcal{C}_{k}$ be the family of the graphs G defined as follows. The vertex set V is a disjoint union D∪X with |D| even and |X|=k|D|. Each component of G[D] is a P ₂ and G[X] is connected of maximum degree at most k. The neighborhoods in X of the vertices of D are all disjoint and of size k. Let $\mathcal{C}=\bigcup_{k=1}^{\infty} \mathcal{C}_{k}$.

Theorem 13

Let G be a connected graph of order n≥2 and maximum degree Δ. Then

$$\gamma_{toc}(G) \ge \gamma_t(G) \ge\frac{n}{\varDelta}. $$

Equalities $\gamma_{toc}(G)= \gamma_{t}(G) = \frac{n}{\varDelta}$ occur if and only if $G\in \mathcal{C}$ or G is a K ₂.

Proof

If Δ=1, then G is a K ₂ and $\gamma_{toc}(G)= \gamma_{t}(G) = 2=\frac {n}{\varDelta}$. Assume now Δ≥2. Let D be a γ _toc (G)-set and X=V∖D. Since D is a total dominating set, d _D (v)≥1, d _X (v)≤Δ−1 for every v∈D, and X⊆N(D). Hence |X|≤(Δ−1)|D| and n≤Δ|D|. Note that we did not use the fact that G[V∖D] is connected. This implies also γ _t (G)≥n/Δ.

If $|D|= \frac{n}{\varDelta}$, then all the inequalities above become equalities. Every vertex v of D has degree 1 in D, thus each component of G[D] is a path P ₂, and degree Δ−1 in X. Since |X|=(Δ−1)|D|, the neighborhoods in X of the vertices of D are disjoint and each vertex of X has one neighbor in D and degree at most Δ−1 in X. Finally, G[X] is connected since D is a TOCD set of G. Hence G is the cycle C ₄, the unique graph of $\mathcal{C}_{1}$, if Δ=2 and belongs to $\mathcal{C}_{\varDelta-1}$ if Δ≥3.

Conversely, let $G\in \mathcal{C}_{k}$ for some integer k≥1. Then Δ(G)=k+1, n=|D|(k+1)=|D|Δ and D is a TOCD set of G of order $\frac {n}{\varDelta}$. Hence $\gamma_{toc}(G)\le \frac{n}{\varDelta}$ and thus $\gamma_{toc}(G)= \frac{n}{\varDelta}$ by the direct part of the theorem. □

3 Graphs of diameter 2

The value of γ _toc (G) may be very large for graphs with large diameter since for the long paths and cycles, γ _toc (G)=n−2. In this section we limit the diameter for the graph and we establish a sharp upper bound on γ _toc (G) in term of the number of vertices for the graphs G for which $\mathrm{diam\,}(G)\le2$ and δ(G)≥3. In order to characterize the extremal graphs, we first describe and study an infinite family of graphs.

Definition 14

A graph G belongs to Family $\mathcal{F}$ if G is obtained by joining a vertex u to every vertex of the disjoint union of a cycles C ₃ and b cycles C ₆ with b=0 and a≥1 or b≥1 and a+2b≥5.

Proposition 15

Let G be a graph of order n obtained by joining a vertex u to every vertex of the disjoint union of a≥0 cycles C ₃ and b≥0 cycles C ₆ with a+b≥1. Then $\gamma_{toc}(G)\le \frac{2n-2}{3}$ with equality if and only if $G\in\mathcal{F}$.

Proof

Case 1 If a+b=1, then G is a wheel and γ _toc (G)=2. If $G\in \mathcal{F}$, then a=1 and b=0, G≃K ₄ and $\gamma_{toc}(G)=2=\frac{2n-2}{3}$. If $G\notin \mathcal{F}$, then a=0 and b=1, n=7 and $\gamma_{toc}(G)=2 <\frac{2n-2}{3}$.

Case 2 If a+b≥2, then n≥7 and u is a cut-vertex of G. Let H ₁,…,H _a+b be the components of G−u. We note that $\gamma_{t}(H_{i})=\frac{2n(H_{i})}{3}$ for 1≤i≤a+b. Let D ₁ be a minimum TOCD set of G containing u and D ₂ a minimum TOCD set of G not containing u. Then γ _toc (G)=min{|D ₁|,|D ₂|}. The set D ₁ contains of all the components of G−u but the largest one and |D ₁|=n−6 if b≥1, |D ₁|=n−3 if b=0. The set D ₂ consists of a minimum total dominating set of each component of G−u and $|D_{2}|=\sum_{i=1}^{a+b}\frac{2n(H_{i})}{3}=\frac {2n-2}{3}$. If b=0, then $G\in \mathcal{F}$ and since n≥7 we have $\gamma_{toc}(G)=\min\{n-3, \frac {2n-2}{3}\}= \frac{2n-2}{3}$. If b≥1 then $\gamma_{toc}(G)=\min\{n-6, \frac{2n-2}{3}\}\le\frac{2n-2}{3}$. Moreover, $\gamma_{toc}(G)=\frac{2n-2}{3}$ if and only if n≥16, i.e., since n=3a+6b+1, if and only if a+2b≥5, in other terms if and only if $G\in\mathcal{F}$. □

Proposition 16

Let G be a graph of order n=7, minimum degree δ=3 and such that γ _toc (G)=4. Then $G\in\mathcal{F}$.

Proof

Let v be a vertex of degree 3, N(v)={v ₁,v ₂,v ₃} and X=V∖N[v]={x,y,z}. Since δ=3, each vertex of X has at least one neighbor in N(v).

Case 1 G[X] is connected. Then G[X] is a path xyz or a cycle C ₃.

Subcase 1.1 Only one vertex of N(v), say v ₁, has a neighbor in X. Then v ₁ is adjacent to each of x,y,z. Since each vertex has degree at least 3, each of X and N(v) induces a C ₃. Therefore $G\in\mathcal{F}$ with u=v ₁, a=2 and b=0.

Subcase 1.2 Exactly two vertices of N(v), say v ₁ and v ₂, have neighbors in X. Then v ₃ is adjacent to v ₁ and v ₂. Without loss of generality, suppose v ₁ adjacent to y. If v ₁ is also adjacent to x and z, then {v,v ₁} is a TOCD set of order two of G. If v ₁ is not adjacent to z (the case v ₁ is not adjacent to x is similar), then v ₂ is adjacent to z and G[X] is a C ₃ since z has degree 3. Then {v ₂,v ₃,z} is a TOCD set of order three of G. This contradicts γ _toc (G)=4.

Subcase 1.3 Each vertex v ₁,v ₂,v ₃ has a neighbor in X.

If there exists no perfect matching between N(v) and X, there are, without loss of generality, two possibilities. Either {v ₁ x,v ₁ y,v ₂ z,v ₃ z}⊆E(G) and {y,z,v ₃} is a TOCD set of order 3 of G. Or {v ₁ x,v ₁ z,v ₂ y,v ₃ y}⊆E(G) and {v,v ₁,z} is a TOCD set of order 3 of G.

Suppose there exists a perfect matching v ₁ x,v ₂ y,v ₃ z between N(v) and X. If G[X] is a cycle, {v,v ₁,x} is a TOCD set of order 3 of G. Suppose xz∉E(G). If v ₃ is adjacent to one of x,y or v ₂, then {x,y,v ₂} is a TOCD set of order 3 of G. Suppose v ₃ is not adjacent to {x,y,v ₂} and similarly v ₁ is not adjacent to {y,z,v ₂}. Since δ=3, v ₁ v ₃,v ₂ x and v ₂ z are in E(G) and {v ₁,v ₂,x} is a TOCD set of order 3 of G. This contradicts γ _toc (G)=4.

Case 2 X is an independent set of G.

Since δ=3, all the edges exist between N(v) and X and {v,v ₁,x} is a TOCD set of order three of G. This contradicts γ _toc (G)=4.

Case 3 G[X] consists of a path xy and an isolated vertex z.

Since δ=3, z is adjacent to v ₁,v ₂,v ₃ and each of x,y has at least two neighbors in N(v). Hence at least one of v ₁,v ₂,v ₃, say v ₁ without loss of generality, is adjacent to x,y and z. Then {v,v ₁} is a TOCD set of order two of G. This contradicts γ _toc (G)=4 and completes the proof. □

Theorem 17

Let G be a graph of order n, $\mathrm{diam\,}(G)\le2$ and δ(G)≥3. Then

$$\gamma_{toc}(G)\le \biggl\lfloor\frac{2n-2}{3}\biggr\rfloor. $$

Furthermore, $\gamma_{toc}(G)= \frac{2n-2}{3}$ if and only if $G \in\mathcal{F}$.

Proof

Let v∈V(G) be a vertex of minimum degree δ, N(v)={v ₁,…,v _δ } and let X=V−N[v]. If X=∅, then V(G)=N[v] and since d(v)=δ we deduce that G is a complete graph. Thus

$$\gamma_{toc}(G)=2\le\biggl\lfloor\frac{2n-2}{3}\biggr\rfloor\quad \mathrm{and}\quad \gamma_{toc}(G)= \frac{2n-2}{3} \quad \mbox{if and~only~if}\ G \simeq K_4 \in \mathcal{F}. $$

Now assume X≠∅. It follows that $\mathrm{diam\,}(G)= 2$ and every vertex of X has at least one neighbor in N(v). Furthermore since d(v)=δ, every vertex v _i with no neighbor in X is adjacent to every vertex of N(v)∖{v _i } and every vertex isolated in G[X] is adjacent to every vertex of N(v). Without loss of generality, we may assume v ₁ is adjacent to some vertex in X. Then {v ₁}∪X is a TOCD set of G and hence γ _toc (G)≤n−δ. If $\delta\ge \frac{n+3}{3}$, then

$$\gamma_{toc}(G)\le n- \frac{n+3}{3} =\frac{2n-3}{3} < \frac{2n-2}{3}. $$

Thus we assume

$$ 3 \le\delta\le\frac{n+2}{3} . $$

(4)

This implies n≥7. If the induced subgraph G[X] is connected, then the total dominating set N[v] is a TOCD set of G and

$$\gamma_{toc}(G)\le \delta+1\le \frac{n+5}{3} \le\frac{2n-2}{3} . $$

If $\gamma_{toc}(G)=\frac{2n-2}{3}$ then n=7, δ=3, γ _toc (G)=4 and $G\in\mathcal{F}$ by Proposition 16.

Let now G ₁,G ₂,…,G _k be the components of G[X] with k≥2. Then for 1≤i≤k,

$$ \bigl|V(G_i)\bigr|\le n-\delta-2 \le n-5. $$

(5)

Case 1 Some v _i , say v ₁, has a private neighbor in X with respect to N(v). Without loss of generality, let G ₁ be the component of G[X] containing a private neighbor x of v ₁ and such that |V(G _i )∖pn(v ₁,N(v))| is minimum, where pn(v ₁,N(v)) is the set of the N(v)-private neighbors of v ₁. Since x is at distance at most two from any other vertex, v ₁ is adjacent to each vertex in $\bigcup_{i=2}^{k}V(G_{i})$ and for 2≤i≤δ, each vertex v _i has a neighbor in V(G ₁)∪{v ₁}. Hence the set {v ₁}∪V(G ₁) is a total dominating set of G.

Subcase 1.1 If V(G ₁)∖pn(v ₁,N(v))≠∅, i.e., each component G _i contains a vertex which is not a (v ₁,N(v))-private neighbor, then {v ₁}∪V(G ₁) is a TOCD set of G and γ _toc (G)≤|V(G ₁)|+1. If n≤9, then by (5),

$$\gamma_{toc}(G)\le n-4 < \frac{2n-2}{3}. $$

Let now n≥10 The set V(G)∖V(G ₁) is another TOCD set of G. Therefore

$$\gamma_{toc}(G)\le \min \bigl\{ \bigl|V(G_1)\bigr|+1, n- \bigl|V(G_1)\bigr| \bigr\}\le \frac{n+1}{2} < \frac{2n-2}{3}. $$

Subcase 1.2 If V(G ₁)∖pn(v ₁,N(v))=∅, i.e., all the vertices of G ₁ are private neighbors of v ₁, then v ₁ dominates G. Therefore any γ _t -set of G−v ₁ is a TOCD set of G. Since δ(G−v ₁)≥2, each component of G−v ₁ has order at least 3 and by Theorem A.1,

$$\gamma_{toc}(G)\le \gamma_t(G-v_1)\le\frac{2n-2}{3}. $$

If $\gamma_{toc}(G)= \frac{2n-2}{3}$, then $\gamma_{t}(G-v_{1})= \frac{2n(G-v_{1})}{3}$. By Theorem A.2 and since δ(G−v ₁)≥2, each component of G−v ₁ is a C ₃ or a C ₆ and G is obtained by joining the vertex v ₁ to each vertex of a disjoint union of cycles C ₃ and C ₆. By Proposition 15, and since $\gamma_{toc}(G)= \frac{2n-2}{3}$, $G\in\mathcal{F}$.

Case 2 For each i, the vertex v _i does not have any private neighbor in X with respect to N(v). Hence each vertex in X has at least two neighbors in N(v). Let T ₁ and T ₂ be respectively the sets of the vertices of the K ₁- and K ₂-components of G[X] and let T ₃=X−(T ₁∪T ₂). Every vertex of T ₁ is adjacent to v _i for 1≤i≤δ. If T ₃≠∅, let D be a γ _t (G[T ₃])-set. By Theorem A, $|D|\le \frac{2|T_{3}|}{3}$. We distinguish three cases.

Subcase 2.1 T ₁∪T ₂≠∅ and if T ₂≠∅ then N(v)⊆N(t) for every vertex t in T ₂. Let S={v,v ₁} if T ₃=∅ and S=D∪{v,v ₁} if T ₃≠∅. In both cases, S is a total dominating set since v ₁ dominates T ₁∪T ₂, and G−S is connected since every vertex of T ₃∖D has a neighbor in N(v)∖{v ₁}. Hence γ _toc (G)≤|S| and

$$\gamma_{toc}(G) \le \frac{2|T_3|}{3}+2 \le \frac{2(n-\delta-2)}{3}+2 = \frac{2n-2\delta+2}{3} \le\frac {2n-4}{3}< \frac{2n-2}{3}. $$

Subcase 2.2 T ₂≠∅ and there exists a vertex t of T ₂ such that |N(t)∩N(v)|<δ. Then deg(t)=δ and without loss of generality, N(t)∩N(v)=N(v)∖{v ₁}. Let t′ be the neighbor of t in X. If some neighbor of t has a private neighbor with respect to N(t), then we are done by exchanging v and t and applying Case 1. Hence we suppose that for 2≤i≤δ, v _i has no private neighbor with respect to N(t). Every vertex x in X∖{t,t′} has at least one neighbor v _i in {v ₂,v ₃,…,v _δ }. Moreover, since x is not a N(t)-private neighbor of v _i and is not adjacent to t′, it has actually at least two neighbors in {v ₂,v ₃,…,v _δ } and thus at least one in {v ₃,…,v _δ }. Let S={v,v ₁,v ₂} if T ₃=∅ and S=D∪{v,v ₁,v ₂} if T ₃≠∅. Every vertex of T ₁∪T ₂ has at least one neighbor in {v ₁,v ₂}. Therefore S is a total dominating set of G. Moreover V(G)∖S is connected in G since t is adjacent to v ₃,v ₄,…,v _δ ,t′ and every vertex of X∖{t,t′} has a neighbor in {v ₃,…,v _δ }. Hence S is a TOCD set of G and

$$\gamma_{toc}(G)\le\frac{2|T_3|}{3}+3\le\frac{2(n-\delta-3)}{3}+3 =\frac {2n-2\delta+3}{3} \le \frac{2n-3}{3}<\frac{2n-2}{3}. $$

Subcase 2.3 T ₁∪T ₂=∅ and thus every component G _i of G[X] has at least three vertices. Let without loss of generality {v ₁,v ₂,…,v _r } be the set of vertices of N(v) with at least one neighbor in X. For i=1,2, we choose a vertex u _i of G _i such that u _i is a support vertex of G _i if δ(G _i )=1, u _i is any vertex of G _i otherwise. Since each of u ₁, u ₂ has at least two neighbors in N(v), there exist two non-adjacent edges between N(v) and {u ₁,u ₂}, say without loss of generality v ₁ u ₁ and v ₂ u ₂. When r≥3, we associate to each vertex v _i with 3≤i≤r one of its neighbors u _i in X. Let H be the subgraph of G defined by V(H)=X∪{v ₁,…,v _r } and E(H)=E(G[X])∪{u ₁ v ₁,…,u _r v _r }. The graph H has order n(H)=n−δ−1+r≤n−1. Let H ₁,…,H _k be the components of H. Each of them consists of a component of G[X] plus possibly some pendant edges and thus has at least three vertices. Moreover, by the choice of u ₁ and u ₂, the components H ₁ and H ₂ are not cycles, 2-coronas, nor 2-coronas minus a pendant edge. By Theorem A, $\gamma_{t}(H_{i})\le\frac{2n(H_{i})-2}{3}$ for 1≤i≤2. Let Y be a γ _t (H)-set not intersecting the set of leaves {v ₁,…,v _r } and thus containing {u ₁,…,u _r }. It follows from Theorem A that

The set S=Y∪{v ₁} is a total dominating set of G since every vertex v _i with r+1≤i≤δ, if any, is adjacent to v ₁. The set V(G)∖S is connected in G since every vertex of X∖Y has at least one neighbor in N(v)∖{v ₁}. Therefore S is a TOCD set of G and $\gamma_{toc}(G)\le|Y|+1\le\frac{2n-3}{3}<\frac {2n-2}{3}$.

The proof of the direct part of the theorem is complete. Conversely, Proposition 15 shows that if $G\in\mathcal{F}$, then $\gamma_{toc}(G)=\frac{2n-2}{3}$. □

Remark 1

The Dutch-windmill graph, $K_{3}^{(m)}$, is a graph which consists of m copies of K ₃ with a vertex in common. Since $\mathrm{diam\,}(K_{3}^{(m)})=2$ and $\gamma_{toc}(K_{3}^{(m)})=n-2$, the graph $K_{3}^{(m)}$ shows that the condition δ(G)≥3 in Theorem 17 cannot be replaced by δ(G)≥2.

4 Complexity issues

We consider the following decision problem.

TOTAL OUTER-CONNECTED DOMINATING SET (TOCDS)

INSTANCE: A graph G=(V,E) and a positive integer k≤|V|.

QUESTION: Does G have a total outer-connected dominating set of cardinality at most k?

Cyman (2007) showed that the similar decision problem for outer-connected dominating set (OCDS) was NP-complete in bipartite graphs by making a reduction from the problem Exact cover by 3-sets. The reader can check in Cyman (2007) that the construction of her proof shows the same result if we replace outer-connected dominating sets by total outer-connected dominating sets. Hence we can state without any new proof:

Theorem 18

TOCDS is NP-complete for bipartite graphs.

Note that the same proof also shows the NP-completeness in bipartite graphs of the decision problem related to the doubly connected dominating number (the dominating set S and V∖S are connected). This result was already established by Cyman et al. (2006) with another construction.