Graphs with no even holes and no sector wheels are the union of two chordal graphs

TL;DR

This paper proves a special case of Sivaraman's conjecture, showing that graphs with no even holes and no sector wheels can be decomposed into two chordal graphs, advancing understanding of graph structure.

Contribution

It confirms Sivaraman's conjecture for graphs without sector wheels, demonstrating a specific structural decomposition into two chordal subgraphs.

Findings

Graphs with no even holes and no sector wheels can be decomposed into two chordal graphs.

The conjecture holds in the special case with no sector wheels.

Provides structural insights into a class of graphs with forbidden induced subgraphs.

Abstract

Sivaraman conjectured that if $G$ is a graph with no induced even cycle then there exist sets $X_1, X_2 \subseteq V(G)$ satisfying $V(G) = X_1 \cup X_2$ such that the induced graphs $G[X_1]$ and $G[X_2]$ are both chordal. We prove this conjecture in the special case where $G$ contains no sector wheel, namely, a pair $(H, w)$ where $H$ is an induced cycle of $G$ and $w$ is a vertex in $V(G) \setminus V(H)$ such that $N(w) \cap H$ is either $V(H)$ or a path with at least three vertices.