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

Abstract

Sivaraman (2020) conjectured that if $G$ is a graph with no induced even cycle then there exist sets $X_1,X_2\subseteq V\left(G\right)$ satisfying $V\left(G\right)=X_1\cup X_2$ such that the induced graphs $G\left[X_1\right]$ and $G\left[X_2\right]$ 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 $V\left(G\right)\setminus V\left(H\right)$ such that $N\left(w\right)\cap H$ is either $V\left(H\right)$ or a path with at least three vertices.