Isometric path partition: a new upper bound and a characterization of some extremal graphs

TL;DR

This paper establishes an upper bound for the isometric path partition number of graphs based on their vertices and matching number, and characterizes the extremal graphs where this bound is tight.

Contribution

It introduces a new upper bound for the isometric path partition number and characterizes extremal graphs achieving equality.

Findings

Proves that $ ext{ipp}(G) \

|V(G)| - u(G)$ for all graphs.

Characterizes extremal graphs where equality holds, involving odd complete blocks and specific conditions on blocks.

Abstract

An $\textit{isometric path}$ is a shortest path between two vertices. An $\textit{isometric path partition}$ (IPP) of a graph $G$ is a set $I$ of vertex-disjoint isometric paths in $G$ that partition the vertices of $G$. The \textit{isometric path partition number} of $G$, denoted by $\text{ipp}(G)$, is the minimum cardinality of an IPP of $G$. In this article, we prove that every graph $G$ satisfies $\text{ipp}(G) \leq |V(G)| - \nu(G)$, where $\nu(G)$ is matching number of $G$. We further prove that a connected graph $G$ is extremal with respect to this upper bound, i.e.\ satisfies $\text{ipp}(G) = |V(G)| - \nu(G)$, if and only if either (i) all blocks of $G$ are odd complete graphs, or (ii) all blocks of $G$ except one are odd complete graphs, and the unique block $B$ of $G$ that is not an odd complete graph is even and satisfy $\text{ipp}(B) = |V(B)| - \nu(B)$. As corollaries of this result, we obtain a full structural characterization of all connected odd graphs that are extremal with respect to our upper bound, as well as of all extremal block graphs.