Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2001.00083

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

Nanao Kita
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.COcs.DM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nanao Kita. 2019-12-31. Constructive Characterization for Bidirected Analogue of Critical Graphs I: Principal Classes of Radials and Semiradials. https://arxiv.org/abs/2001.00083

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗