Vertebrate interval graphs

TL;DR

This paper introduces polynomial-time algorithms for partitioning vertebrate interval graphs into subgraphs with bounded claw number, including proper interval graph partitions, advancing understanding of their structural properties.

Contribution

It presents the first polynomial-time algorithms for vertex partition problems in vertebrate interval graphs with bounded claw number.

Findings

Polynomial-time algorithm for vertebrate interval graphs with claw number at most v

Decides partition into two proper interval graphs when v=2

Characterizes vertebrate interval graphs in terms of independent sets and maximal cliques

Abstract

A vertebrate interval graph is an interval graph in which the maximum size of a set of independent vertices equals the number of maximal cliques. For any fixed $v \ge 1$, there is a polynomial-time algorithm for deciding whether a vertebrate interval graph admits a vertex partition into two induced subgraphs with claw number at most $v$. In particular, when $v = 2$, whether a vertebrate interval graph can be partitioned into two proper interval graphs can be decided in polynomial time.