Sum-perfect graphs

TL;DR

This paper introduces sum-perfect graphs, a new class characterized by a specific inequality involving independence and clique numbers, and provides a forbidden subgraph characterization for these graphs.

Contribution

The paper defines sum-perfect graphs and establishes a forbidden induced subgraph characterization using a set of 27 graphs.

Findings

Sum-perfect graphs are characterized by 27 forbidden induced subgraphs.

A new class of graphs related to perfect graphs is introduced.

The characterization provides a basis for further structural and algorithmic studies.

Abstract

Inspired by a famous characterization of perfect graphs due to Lov\'{a}sz, we define a graph $G$ to be sum-perfect if for every induced subgraph $H$ of $G$, $\alpha(H) + \omega(H) \geq |V(H)|$. (Here $\alpha$ and $\omega$ denote the stability number and clique number, respectively.) We give a set of $27$ graphs and we prove that a graph $G$ is sum-perfect if and only if $G$ does not contain any of the graphs in the set as an induced subgraph.