Minimum $T$-Joins and Signed-Circuit Covering

TL;DR

This paper investigates bounds on minimum T-joins in graphs and applies these results to establish length bounds for signed-circuit covers in flow-admissible signed graphs, with specific improvements for 2-edge-connected cases.

Contribution

It introduces a new upper bound on the size of minimum T-joins based on the maximum bidegeless subgraph, and applies this to derive length bounds for signed-circuit covers in signed graphs.

Findings

Minimum T-joins have at most |E(G)| - 0.5|E(Ĝ)| edges.

Flow-admissible signed graphs have signed-circuit covers with length ≤ (19/6)|E(G)|.

2-edge-connected signed graphs with even negativeness have covers with length ≤ (8/3)|E(G)|.

Abstract

Let $G$ be a graph and $T$ be a vertex subset of $G$ with even cardinality. A $T$-join of $G$ is a subset $J$ of edges such that a vertex of $G$ is incident with an odd number of edges in $J$ if and only if the vertex belongs to $T$. Minimum $T$-joins have many applications in combinatorial optimizations. In this paper, we show that a minimum $T$-join of a connected graph $G$ has at most $|E(G)|-\frac 1 2 |E(\widehat{\, G\,})|$ edges where $\widehat{\,G\,}$ is the maximum bidegeless subgraph of $G$. Further, we are able to use this result to show that every flow-admissible signed graph $(G,\sigma)$ has a signed-circuit cover with length at most $\frac{19} 6 |E(G)|$. Particularly, a 2-edge-connected signed graph $(G,\sigma)$ with even negativeness has a signed-circuit cover with length at most $\frac 8 3 |E(G)|$.