# A characterization of claw-free CIS graphs and new results on the order of CIS graphs

## TL;DR

This paper provides a structural characterization of claw-free CIS graphs, introduces an efficient recognition method, and addresses a question about the order of CIS graphs, showing it is bounded in claw-free cases.

## Contribution

It characterizes claw-free CIS graphs through a new structural property and confirms the boundedness of their order relative to clique and stability numbers.

## Key findings

- Characterization of graphs where every maximal matching saturates vertices of degree at least two.
- Efficient test for claw-free CIS graphs based on the structural characterization.
- Proof that the order of CIS graphs is bounded by the product of clique and stability numbers in the claw-free case.

## Abstract

A graph is CIS if every maximal clique interesects every maximal stable set. Currently, no good characterization or recognition algorithm for the CIS graphs is known. We characterize graphs in which every maximal matching saturates all vertices of degree at least two and use this result to give a structural, efficiently testable characterization of claw-free CIS graphs. We answer in the negative a question of Dobson, Hujdurovi\'c, Milani\v{c}, and Verret [Vertex-transitive CIS graphs, European J. Combin. 44 (2015) 87-98] asking whether the number of vertices of every CIS graph is bounded from above by the product of its clique and stability numbers. On the positive side, we show that the question of Dobson et al. has an affirmative answer in the case of claw-free graphs.