Local $3$-Monophonic Convexity

Abstract

Let $G$ be a connected graph and let $U$ be a set of vertices in $G$. A $minimalU-tree$ is a subtree $T$ of $G$ that contains $U$ and has the property that every vertex of $V(T)–U$ is a cut-vertex of $⟨V(T)⟩$. The $monophonicinterval$ of $U$ is the collection of all vertices of $G$ that lie on some minimal $U$-tree. A set $S$ of vertices of $G$ is $m_k$-$convex$ if it contains the monophonic interval of every $k$-subset $U$ of vertices of $S$. Thus $S$ is $m_2$-convex if and only if it is $m$-convex.

In this paper, we consider three local convexity properties with respect to $m_3$-convexity and characterize the graphs having either property.