Induced subgraphs of graphs with large deficiency

TL;DR

This paper explores the structure of graphs with large deficiency and bounded local independence number, identifying special subgraphs called bones, and provides bounds and algorithms related to maximum matchings.

Contribution

It introduces the concept of bones in graphs with large deficiency and bounded local independence number, establishing extremal bounds and an algorithm for large matchings.

Findings

Identifies bones as key subgraphs in graphs with large deficiency.

Establishes exact bounds on deficiency for graphs excluding certain bones.

Provides an algorithm to construct large matchings and bound deficiency.

Abstract

The deficiency of a graph $G$, denoted by $\kd(G)$, is the number of vertices not saturated by a maximum matching. A bone $B_i$ is the tree obtained by attaching two pendent edges to each of the end vertices of a path $P_{i}$. The local independence number of $G$, denoted by $\alpha_l(G)$, is defines as the maximum integer $t$ such that $G$ contains an induced star $K_{1,t}$. Motivated by the seminal works of Scott and Seymour~(2016), Chudnovsky et al. (2017, 2020) on finding special types of holes in graphs with large chromatic number and bounded clique number, we establish an analog result by finding special types of bones in graphs with large deficiency and bounded local independence number. Fujita et al. (2006) proved that $\kd(G)\le n-2$ if $G$ is a connected graph with $\alpha_l(G)<n$ and containing no bones. We further establish exact extremal deficiency bounds for connected graphs with bounded local independence number that exclude specific bone configurations. An algorithm that constructs large matchings and establishes an upper bound on the deficiency is also provided.