Constructive Characterization for Bidirected Analogue of Critical Graphs I: Principal Classes of Radials and Semiradials

TL;DR

This paper introduces constructive characterizations for five principal classes of radials and semiradials in bidirected graphs, generalizing concepts from matching theory and digraphs, to facilitate their analysis and decomposition.

Contribution

It provides the first constructive characterizations for key classes of radials and semiradials in bidirected graphs, expanding the theoretical framework for their analysis.

Findings

Defined new classes of radials and semiradials in bidirected graphs.

Provided constructive characterizations for five principal classes.

Established groundwork for strong component decomposition of bidirected graphs.

Abstract

This paper is the first from serial papers that provide constructive characterizations for classes of bidirected graphs known as radials and semiradials. In this paper, we provide constructive characterizations for five principle classes of radials and semiradials to be used for characterizing general radials and semiradials. A bidirected graph is a graph in which each end of each edge has a sign $+$ or $-$. Bidirected graphs are a common generalization of digraphs and signed graphs. We define a new concept of radials as a generalization of a classical concept in matching theory, critical graphs. Radials are also a generalization of a class of digraphs known as flowgraphs. We also define semiradials, which are a relaxed concept of radials. We further define special classes of radials and semiradials, that is, absolute semiradials, strong and almost strong radials, linear semiradials, and sublinear radials. We provide constructive characterizations for these five classes of bidirected graphs. Our serial papers are a part of a series of works that establish the strong component decomposition for bidirected graphs.