Claw-freeness, 3-homogeneous subsets of a graph and a reconstruction problem

TL;DR

This paper characterizes graphs whose structure and complement are both claw-free, describes the hypergraph of 3-vertex subsets inducing cliques or independent sets, and explores implications for graph reconstruction up to complementation.

Contribution

It provides a detailed classification of a special class of graphs and links this to hypergraph representations and a graph reconstruction problem.

Findings

Characterization of graphs with both the graph and its complement being claw-free.

Description of the hypergraph of 3-vertex subsets related to these graphs.

Insights into the graph reconstruction problem up to complementation.

Abstract

We describe $Forb\{K_{1,3}, \overline {K_{1,3}}\}$, the class of graphs $G$ such that $G$ and its complement $ \overline{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths or isolated vertices, and of the complements of these graphs. Considering the hypergraph ${\mathcal H} ^{(3)}(G)$ made of the 3-element subsets of the vertex set of a graph $G$ on which $G$ induces a clique or an independent subset, we deduce from above a description of the Boolean sum $G\dot{+}G'$ of two graphs $G$ and $G'$ giving the same hypergraph. We indicate the role of this latter description in a reconstruction problem of graphs up to complementation.