Graphs whose indecomposability graph is 2-covered

TL;DR

This paper characterizes indecomposable graphs whose indecomposability graph has a vertex cover of size two, providing insights into the structure of such graphs and their intervals.

Contribution

It introduces a characterization of indecomposable graphs with indecomposability graphs that are 2-covered, advancing understanding of graph decomposability and interval structures.

Findings

Characterization of indecomposable graphs with 2-covered indecomposability graphs

Identification of structural properties leading to 2-covered indecomposability graphs

Extension of graph decomposition theory to new classes of graphs

Abstract

Given a graph $G=(V,E)$, a subset $X$ of $V$ is an interval of $G$ provided that for any $a, b\in X$ and $ x\in V \setminus X$, $\{a,x\}\in E$ if and only if $\{b,x\}\in E$. For example, $\emptyset$, $\{x\}(x\in V)$ and $V$ are intervals of $G$, called trivial intervals. A graph whose intervals are trivial is indecomposable; otherwise, it is decomposable. According to Ille, the indecomposability graph of an undirected indecomposable graph $G$ is the graph $\mathbb I(G)$ whose vertices are those of $G$ and edges are the unordered pairs of distinct vertices $\{x,y\}$ such that the induced subgraph $G[V \setminus \{x,y\}]$ is indecomposable. We characterize the indecomposable graphs $G$ whose $\mathbb I(G)$ admits a vertex cover of size 2.