Les graphes (-1)-critiques

TL;DR

This paper characterizes a new class of indecomposable graphs called (-1)-critical graphs, which have exactly one non-critical vertex, expanding the understanding of graph decomposability and criticality.

Contribution

It provides a complete characterization of (-1)-critical indecomposable graphs, answering a recent open question about graphs with a single non-critical vertex.

Findings

Characterization of (-1)-critical graphs

Extension of critical vertex theory in indecomposable graphs

Addresses an open problem in graph decomposability

Abstract

Given a (directed) graph G=(V,A), a subset X of V is an interval of G provided that for any a, b\in X and x\in V-X, (a,x)\in A if and only if (b,x)\in A and (x,a)\in A if and only if (x,b)\in A. For example, \emptyset, \{x\} (x \in V) and V are intervals of G, called trivial intervals. A graph, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A vertex x of an indecomposable graph is critical if G-x is decomposable. In 1993, J.H. Schmerl and W.T. Trotter characterized the indecomposable graphs, all the vertices of which are critical, called critical graphs. In this article, we characterize the indecomposable graphs which admit a single non critical vertex, that we call (-1)-critical graphs.} This gives an answer to a question asked by Y. Boudabbous and P. Ille in a recent article studying the critical vertices in an indecomposable graph.