On path sequences of graphs

TL;DR

This paper studies the sequence of minimum vertex covers for all path lengths in a graph, providing conditions for sequence entries and classifying possible sequences for small graphs.

Contribution

It establishes necessary and sufficient conditions for specific entries in the path sequence and classifies all possible sequences for small connected graphs.

Findings

Characterization of when two integers appear at fixed positions in the sequence

Complete classification of path sequences for small connected graphs

Conditions for the realizability of path sequences

Abstract

A subset $S$ of vertices of a graph $G=(V,E)$ is called a $k$-path vertex cover if every path on $k$ vertices in $G$ contains at least one vertex from $S$. Denote by $\psi_k(G)$ the minimum cardinality of a $k$-path vertex cover in $G$ and form a sequence $\psi(G)=(\psi_1(G),\psi_2(G),\ldots,\psi_{|V|}(G))$, called the path sequence of $G$. In this paper we prove necessary and sufficient conditions for two integers to appear on fixed positions in $\psi(G)$. A complete list of all possible path sequences (with multiplicities) for small connected graphs is also given.