New Vertex Ordering Characterizations of Circular-Arc Bigraphs

TL;DR

The paper addresses the problem of characterizing circular-arc bigraphs, a bipartite extension of circular-arc graphs, by introducing order-based methods. It presents two main characterizations: total-circular ordering and bi-circular ordering, establishing equivalence with circular-arc bigraphs, and a forbidden-pattern criterion analogous to Hell-Huang's approach. A key contribution is demonstrating that circular-arc bigraphs can be recognized via these orderings and patterns, paralleling results for interval and circular-arc graphs. The work lays groundwork for future efficient recognition (and certifying recognition) of circular-arc bigraphs and connects to established results on circular-arc graphs.

Abstract

In this article, we present two new characterizations of circular-arc bigraphs based on their vertex ordering. Also, we provide a characterization of circular-arc bigraphs in terms of forbidden patterns with respect to a particular ordering of their vertices.

Figures (10)

• Figure 1: A circular-arc bigraph where a total-circular ordering of the vertices is: $x_1$, $y_2$, $y_3$, $x_4$, $y_5$, $x_6$, $y_7$, $x_8$.
• Figure 2: A bigraph having an ordering of its vertices: $y_1$, $x_2$, $x_3$, $y_4$, $y_5$, $x_6$, $y_7$, $y_8$, $x_9$, $y_{10}$. It is a bi-circular ordering as shown in the next figure.
• Figure 3: The biadjacency matrix of the bigraph in Figure 2, where the rows and columns are arranged according to the increasing order of their indices, and corresponding $W_i$'s and $W_j$'s.
• Figure 4: Forbidden pattern.
• Figure 5: Forbidden patterns.

Theorems & Definitions (8)

• Definition 1
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3: hell
• Theorem 4
• proof