Vertex ordering characterizations of interval r-graphs

TL;DR

This work addresses recognizing and characterizing interval $r$-graphs via vertex orderings. It develops two complementary orderings—generalized interval ordering and $r$-interval ordering—and proves that an $r$-partite graph is an interval $r$-graph if and only if its vertices admit one of these orderings, connecting order-based structure to interval representations. The paper also establishes forbidden-pattern characterizations in terms of vertex orderings, extending concepts from interval bigraphs to the $r ext{-graph}$ setting and introducing the notion of almost consecutive ones to capture adjacency in the matrix. These results provide structural and algorithmic tools for recognizing interval $r$-graphs and pave the way for future work on circular-arc generalizations and finite forbidden-pattern sets.

Abstract

An r-partite graph is an interval r-graph if corresponding to each vertex we can assign an interval of the real line such that two vertices u and v of different partite sets are adjacent if and only if their corresponding intervals intersect. In this paper, we provide two vertex-ordering characterizations of interval r-graphs and identify forbidden patterns for interval r-graphs in terms of specific orderings of their vertices.

Figures (8)

• Figure 1: Left end point of $I_{v_i}$ is contained in $I_{v_j}$.
• Figure 2: $I_{v_i}\cap I_{v_j}\neq\emptyset$, where $i>j$.
• Figure 3: $I_{v_i}\cap I_{v_j}\neq\emptyset$, where $i<j$.
• Figure 4: A $3$-partite graph.
• Figure 5: The adjacency matrix of the $3$ partite graph in Figure 2, where the rows and columns are arranged according to the increasing order of their indices, and corresponding $\mathcal{R}_i$'s and $\mathcal{C}_j$'s are shown in the elliptic regions. And its interval representation is as follows: Red: $I_{v_1}=[1,3]$, $I_{v_6}=[6,9]$, $I_{v_{10}}=[10,10]$ Green: $I_{v_2}=[3,4]$, $I_{v_8}=[8,10]$, $I_{v_9}=[9,9]$ Black: $I_{v_3}=[3,3]$, $I_{v_4}=[4,6]$, $I_{v_5}=[5,8]$, $I_{v_7}=[7,9]$.

Theorems & Definitions (10)

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