Positive matching decompositions of graphs

TL;DR

This paper introduces positive matching decompositions (pmds) of graphs, characterizes them via alternating closed walks, and explores their properties across various graph classes, linking them to algebraic graph invariants.

Contribution

It provides a new characterization of pmds using alternating closed walks and applies it to diverse graph classes, advancing understanding of their algebraic properties.

Findings

Characterization of pmds via alternating closed walks

Application to complete multipartite, bipartite, cacti, and Petersen graphs

Reduction of pmd computation to maximum pendant-free subgraph

Abstract

A matching $M$ in a graph $\Gamma$ is positive if $\Gamma$ has a vertex-labeling such that $M$ coincides with the set of edges with positive weights. A positive matching decomposition (pmd) of $\Gamma$ is an edge-partition $M_1,\ldots,M_p$ of $\Gamma$ such that $M_i$ is a positive matching in $\Gamma-M_1\cup\cdots\cup M_{i-1}$, for $i=1,\ldots,p$. The pmds of graphs are used to study algebraic properties of the Lov\'{a}sz-Saks-Schrijver ideals arising from orthogonal representations of graphs. We give a characterization of pmds of graphs in terms of alternating closed walks and apply it to study pmds of various classes of graphs including complete multipartite graphs, (regular) bipartite graphs, cacti, generalized Petersen graphs, etc. We further show that computation of pmds of a graph can be reduced to that of its maximum pendant-free subgraph.