A Recognition Algorithm and Some Optimization Problems on Weakly Quasi-Threshold Graphs

Abstract

Graph theory provides algorithms and tools to handle models for important applications in medicine, such as drug design, diagnosis, validation of graph-theoretical methods for pattern identification in public health datasets. In this chapter we characterize weakly quasi-threshold graphs using the weakly decomposition, determine: density and stability number for weakly quasi-threshold graphs.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

The well-known class of cographs is recursively defined by using the graph operations of ‘union’ and ‘join’ [1]. Bapat et al. [2], introduced a proper subclass of cographs, namely the class of weakly quasi-threshold graphs, by restricting the join operation. The class of cographs coincides with the class of graphs having no induced $P_4$ [3]. Trivially-perfect graphs, also known as quasi-threshold graphs, are characterized as the subclass of cographs having no induced $C_4$, that is, such graphs are $\{P_4,C_4 \}$-free graphs, and are recognized in linear time [4, 5]. Another subclass of cographs are the $\{P_4,C_4, 2K_2\}$-free graphs known as threshold graphs, for which there are several linear-time recognition algorithms [4, 5]. Every threshold graph is trivially-perfect but the converse is not true.

When searching for recognition algorithms, frequently appears a type of partition for the set of vertices in three classes $A,B,C$, which we call a weakly decomposition, such that: $A$ induces a connected subgraph, $C$ is totally adjacent to $B$, while $C$ and $A$ are totally nonadjacent.

The structure of the chapter is the following. In Sect. 2 we present the notations to be used, in Sect. 3 we give the notion of weakly decomposition and in Sect. 4 we give a recognition algorithm and determine the clique number, the stability number on weakly quasi-threshold graphs.

2 General Notations

Throughout this chapter, $G=(V,E)$ is a connected, finite and undirected graph, without loops and multiple edges [6], having $V=V(G)$ as the vertex set and $E=E(G)$ as the set of edges. $\overline{G}$ is the complement of $G$. If $U\subseteq V$, by $G(U)$ we denote the subgraph of $G$ induced by $U$. By $G-X$ we mean the subgraph $G(V-X)$, whenever $X\subseteq V$, but we simply write $G-v$, when $X=\{v\}$. If $e=xy$ is an edge of a graph $G$, then $x$ and $y$ are adjacent, while $x$ and $e$ are incident, as are $y$ and $e$. If $xy\in E$, we also use $x\sim y$, and $x\,{\nsim}\,y$ whenever $x,y$ are not adjacent in $G$. A vertex $z\in V$ distinguishes the non-adjacent vertices $x,y\in V$ if $zx\in E$ and $zy\not \in E$. If $A,B\subset V$ are disjoint and $ab\in E$ for every $a\in A$ and $b\in B$, we say that $A,B$ are totally adjacent and we denote by $A\sim B$, while by $A\,{\nsim}\,B$ we mean that no edge of $G$ joins some vertex of $A$ to a vertex from $B$ and, in this case, we say that $A$ and $B$ are non-adjacent.

The neighbourhood of the vertex $v\in V$ is the set $N_{G}(v)=\{u\in V:uv\in E\}$, while $N_{G}[v]=N_{G}(v)\cup \{v\}$; we simply write $N(v)$ and $N[v]$, when $G$ appears clearly from the context. The neighbourhood of the vertex $v$ in the complement of $G$ will be denoted by $\overline{N}(v)$.

The neighbourhood of $S\subset V$ is the set $N(S)=\cup _{v\in S}N(v)-S$ and $N[S]=S\cup N(S)$. A clique is a subset $Q$ of $V$ with the property that $G(Q)$ is complete. The clique number or density of $G$, denoted by $\omega (G)$, is the size of the maximum clique. A clique cover is a partition of the vertices set such that each part is a clique. $\theta (G)$ is the size of a smallest possible clique cover of $G$; it is called the $clique$ $cover$ $number$ of $G$. A stable set is a subset $X$ of vertices where every two vertices are not adjacent. $\alpha (G)$ is the number of vertices is a stable set o maximum cardinality; it is called the $stability$ $number$ of $G$. $\chi (G) = \omega \overline{(}G)$ and it is called $chromatic$ $number$.

By $P_{n}$, $C_{n}$, $K_{n}$ we mean a chordless path on $n\ge 3$ vertices, a chordless cycle on $n\ge 3$ vertices, and a complete graph on $n\ge 1$ vertices, respectively.

A graph is called $cograph$ if it does not contain $P_4$ as an induced subgraph.

Let $\fancyscript{F}$ denote a family of graphs. A graph $G$ is called $\fancyscript{F}$-free if none of its subgraphs is in $F$.

3 Preliminary Results

3.1 Weakly Decomposition

At first, we recall the notions of weakly component and weakly decomposition.

Definition 1

[7–9] A set $A\subset V(G)$ is called a weakly set of the graph $G$ if $N_{G}(A)\not =V(G)-A$ and $G(A)$ is connected. If $A$ is a weakly set, maximal with respect to set inclusion, then $G(A)$ is called a weakly component. For simplicity, the weakly component $G(A)$ will be denoted with $A$.

Definition 2

[7–9] Let $G=(V,E)$ be a connected and non-complete graph. If $A$ is a weakly set, then the partition $\{A,N(A),V-A\cup N(A)\}$ is called a weakly decomposition of $G$ with respect to $A$.

Below we remind a characterization of the weakly decomposition of a graph.

The name of "weakly component" is justified by the following result.

Theorem 1

[8–10] Every connected and non-complete graph $G=(V,E)$ admits a weakly component $A$ such that $G(V-A)=G(N(A))+G(\overline{N}(A))$.

Theorem 2

[8, 9] Let $G=(V,E)$ be a connected and non-complete graph and $A\subset V$. Then $A$ is a weakly component of $G$ if and only if $G(A)$ is connected and $N(A)\sim \overline{N}(A)$.

The next result, that follows from Theorem 1, ensures the existence of a weakly decomposition in a connected and non-complete graph.

Corollary 1

If $G=(V,E)$ is a connected and non-complete graph, then $V$ admits a weakly decomposition $(A,B,C)$, such that $G(A)$ is a weakly component and $G(V-A)=G(B)+G(C)$.

Theorem 2 provides an $O(n+m)$ algorithm for building a weakly decomposition for a non-complete and connected graph.

In [7] we give:

Let $G=(V,E)$ be a connected graph with at least two nonadjacent vertices and (A,N,R) a weakly decomposition, with A the weakly component. G is a $P_4$-free graph if and only if:

• (1) $A \sim N \sim R$;

• (2) G(A), G(N), G(R) are $P_4$-free graph.

3.2 Weakly Quasi-Threshold Graphs

In this subsection we remind some results on weakly quasi-threshold graphs.

A $cograph$ which is $C_4$-free is called a $quasi$-$threshold$ graph.

In [2] we study the class of weakly quasi-threshold graphs that are obtained from a vertex by recursively applying the operations (1) adding a new isolated vertex, (2) adding a new vertex and making it adjacent to all old vertices, (3) disjoint union of two old graphs, and (4) adding a new vertex an making it adjacent to all neighbours of an old vertex.

Let $G=(V,E)$ be a graph. Define a relation on $V$ [2] as follows: Let $u, v \in V$. Then $u \equiv v$ if $N(u) = N(v)$. We observe that $\equiv$ is an equivalence relation and the equivalence classes are stable sets in $G$.

Let $G$ be a graph with $Q_1, ..., Q_k$ as the equivalence classes under the relation $\equiv$. For each $i=1, ..., k$ choose a vertex $u_i \in Q_i$. We call the subgraph $\widetilde{G}$ of $G$ induced by $u_1, ..., u_k$ as a subgraph of representatives of $G$.

Let $G$ be a graph. Then $G$ is weakly quasi-threshold [2] if an only if a subgraph of representatives is quasi-threshold.

Let $G=(V,E)$ be a connected graph. Then the following are equivalent [2]:

• (1) $G$ is a weakly quasi-threshold

• (2) $G$ ia a $P_4$-free and there is no induced $C_4 = [v_1, v_2, v_3, v_4 ]$ with $N(v_1) \not = N(v_3)$ and $N(v_2) \not = N(v_4)$.

A graph $G$ is weakly quasi-threshold [11] if and only if $G$ does not contain any $P_4$ or $co-(2P_3)$ as induced subgraphs.

4 New Results on Threshold Graphs

4.1 Characterization of a Weakly Quasi-Threshold Graph Using the Weakly Decomposition

In this paragraph we give a new characterization of weakly quasi-threshold graphs using the weakly decomposition.

Theorem 3

Let $G=(V,E)$ be a connected graph with at least two nonadjacent vertices and (A, N, R) a weakly decomposition, with A the weakly component. G is a weakly quasi-threshold graph if and only if:

• (1) $A \sim N \sim R$;

• (2) $G{\rm{(}}N{\rm{)}}$ is $\overline{P}_3$-free graph;

• (3) $G(A \cup N)$, $G(N \cup R)$ are weakly quasi-threshold graphs.

Proof

Let $G=(V,E)$ be a connected, uncomplete graph and $(A,N,R)$ a weakly decomposition of $G$, with $G(A)$ as the weakly component.

At first, we assume that $G$ is weakly quasi-threshold. Then $G$ is $P_4$-free. So, $A \sim N \sim R$. Because $G$ is weakly quasi-threshold graph it follows that $G(A \cup N)$, $G(N \cup R)$ are weakly quasi-threshold graphs. We suppose that $G(N)$ contain $\overline{P}_3 = (\{a,b,c\}, \{ac \})$ as induced subgraph. Because $G(A)$ is connected $\exists x, y \in A$ such that $xy \in E$. Because $A\,{\nsim}\,R$, $\forall z \in R$, $G( \{ x, y, z \} ) \simeq \overline{P}_3$. Because $N \sim A \cup R$, $G( \{ a,y,a,b,c,z \}) \simeq co-(2P_3)$, in contradicting with $G$ is weakly quasi-threshold graph.

Conversely, we suppose that (1), (2) and (3) hold. From (3), $G(A)$, $G(N)$, $G(R)$ are $P_4$-free. Because (1) hold, $G$ is $P_4$-free. $G(A)$, $G(N)$, $G(R)$ are $\{co-(2P_3)\}$-free because (3) hold. $G(A \cup R)$ is $\{ co - (2P_3)\}$-free because $A\,{\nsim}\,R$ and $\{ co-(2P_3)\}$ is connected. We suppose that $G$ contain $H= \{ co-(2P_3)\}$ as induced subgraph such that $V(H) \cap A \not = \emptyset$, $V(H) \cap N \not = \emptyset$ and $V(H) \cap R \not = \emptyset$. Because (1) hold, $N \sim (A \cup R)$. The unique $S \subset V$ totally adjacent with $V(H) -S$, ($S \sim V(H)-S$), is $S$ with $S = V(\overline{P}_3)$. Then $G(N)$ contain $\overline{P}_3$ as induced subgraph, contradicting (2). So, $G$ is $\{co-(2P_3)\}$-free. So, $G$ is weakly quasi- threshold graph.

4.2 Determination of Clique Number and Stability Number for a Weakly Quasi-Threshold Graph

In this paragraph we determine the stability number and the clique number for weakly quasi-threshold graphs.

Proposition 1

If $G=(V,E)$ is a connected graph with at least two nonadjacent vertices and (A, N, R) a weakly decomposition with A the weakly component then

$$\begin{aligned} \alpha (G) = max \{ \alpha (G(A)) + \alpha (G( \overline{N}(A))), \alpha (G(A \cup N(A))) \}. \end{aligned}$$

Proof

Indeed, every stable set of maximum cardinality either intersects $\overline{N}(A)$ and in this case the cardinal is $\alpha (G(A)) + \alpha (G( \overline{N}(A)))$ or it does not intersect $\overline{N}(A)$ and has the cardinal $\alpha (G(A \cup N(A)))$.

Theorem 4

Let $G=(V,E)$ be connected with at least two non-adjacent vertices and (A, N, R) a weakly decomposition with A the weakly component. If G is a weakly quasi-threshold graph then

$$\begin{aligned} \alpha (G) = \alpha (G(A)) + max \{\alpha (G(N)), \alpha (G(R))\} \end{aligned}$$

and

$$\begin{aligned} \omega (G) = \omega (G(N)) + max \{\omega (G(A)), \omega (G(R))\}. \end{aligned}$$

Proof

Because $A \sim N$, from Proposition 1, it follows that

$$\begin{aligned} \alpha (G) = \alpha (G(A)) + max \{\alpha (G(N)), \alpha (G(R))\}. \end{aligned}$$

Because $A \sim N \sim R$, it follows that

$$\begin{aligned} \omega (G) = \omega (G(N)) + max \{\omega (G(A)), \omega (G(R))\}. \end{aligned}$$

5 Conclusions and Future Work

In this chapter we characterize weakly quasi-threshold graphs using the weakly decomposition, determine: density and stability number for weakly quasi-threshold graphs. Our future work concerns we give some applications of weakly quasi-threshold graphs including the medicine. Also we will explore the connection of weakly quasi-threshold graphs with the intelligent systems.