A Characterization of $P_6$-Free Irredundance Perfect Graphs

Vadim Zverovich; Pavel Skums; Lutz Volkmann

A Characterization of $P_6$-Free Irredundance Perfect Graphs

Vadim Zverovich, Pavel Skums, Lutz Volkmann

Abstract

Let $ir(G)$ and $γ(G)$ be the irredundance number and the domination number of a graph G, respectively. A graph G is called irredundance perfect if ir(H) = $γ(H)$ for every induced subgraph $H$ of $G$. The subclass of $P_6$-free irredundance perfect graphs has been studied extensively. In this paper, we present a characterization of this graph class in terms of eleven forbidden induced subgraphs.

A Characterization of $P_6$-Free Irredundance Perfect Graphs

Abstract

Let

and

be the irredundance number and the domination number of a graph G, respectively. A graph G is called irredundance perfect if ir(H) =

for every induced subgraph

. The subclass of

-free irredundance perfect graphs has been studied extensively. In this paper, we present a characterization of this graph class in terms of eleven forbidden induced subgraphs.

Paper Structure (16 theorems, 15 equations, 2 figures)

This paper contains 16 theorems, 15 equations, 2 figures.

Key Result

Proposition 1

Let $X$ be an irredundant set of $G$, and $U=V(G)-N[X]$. The set $X$ is a maximal irredundant set if and only if for any $v\in N[U]$, the vertex $v$ dominates $PN(x,X)$ for some vertex $x\in X$.

Figures (2)

Figure 1: Graphs $G_1-G_5$.
Figure 2: Graphs $F_1-F_{11}$. Dotted lines denote additional edges that should be added to produce the next graph in the corresponding list.

Theorems & Definitions (24)

Proposition 1: Vol4
Theorem 1: Bollobás and Cockayne Bol
Theorem 2: Favaron Fav
Conjecture 1: Favaron Fav, see also Cha
Conjecture 2: Faudree, Favaron and Li Fau
Conjecture 3: Puech Pue
Theorem 3
Lemma 1
Lemma 2
Lemma 3
...and 14 more