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Interval H-graphs : Recognition and forbidden obstructions

Haiko Müller, Arash Rafiey

TL;DR

The paper addresses recognizing interval $H$-graphs, a color-aware generalization of interval graphs, by encoding colored forbidden patterns into a pair-digraph $G^+$. It establishes an ordering characterization and delivers a polynomial-time recognition algorithm for interval $k$-graphs with a fixed partition, with guidance for extending to general $H$. The approach hinges on a reachability/envelope framework ($N^+[R]$, $N^*[R]$) and obstruction structures such as exobicliques, ensuring correctness through circuit-free conditions. The work lays groundwork for recognizing interval $H$-graphs in polynomial time and raises open questions about partition-free recognition and comprehensive obstruction enumeration.

Abstract

We introduce the class of interval $H$-graphs, which is the generalization of interval graphs, particularly interval bigraphs. For a fixed graph $H$ with vertices $a_1,a_2,\dots,a_k$, we say that an input graph $G$ with given partition $V_1,\dots,V_k$ of its vertices is an interval $H$-graph if each vertex $v \in G$ can be represented by an interval $I_v$ from a real line so that $u \in V_i$ and $v \in V_j$ are adjacent if and only if $a_ia_j$ is an edge of $H$ and intervals $I_u$ and $I_v$ intersect. $G$ is called interval $k$-graph if $H$ is a complete graph on $k$ vertices. and interval bigraph when $k=2$. We study the ordering characterization and forbidden obstructions of interval $k$-graphs and present a polynomial-time recognition algorithm for them. Additionally, we discuss how this algorithm can be extended to recognize general interval $H$-graphs. Special cases of interval $k$-graphs, particularly comparability interval $k$-graphs, were previously studied in [2], where the complexity interval $k$-graph recognition was posed as an open problem.

Interval H-graphs : Recognition and forbidden obstructions

TL;DR

The paper addresses recognizing interval -graphs, a color-aware generalization of interval graphs, by encoding colored forbidden patterns into a pair-digraph . It establishes an ordering characterization and delivers a polynomial-time recognition algorithm for interval -graphs with a fixed partition, with guidance for extending to general . The approach hinges on a reachability/envelope framework (, ) and obstruction structures such as exobicliques, ensuring correctness through circuit-free conditions. The work lays groundwork for recognizing interval -graphs in polynomial time and raises open questions about partition-free recognition and comprehensive obstruction enumeration.

Abstract

We introduce the class of interval -graphs, which is the generalization of interval graphs, particularly interval bigraphs. For a fixed graph with vertices , we say that an input graph with given partition of its vertices is an interval -graph if each vertex can be represented by an interval from a real line so that and are adjacent if and only if is an edge of and intervals and intersect. is called interval -graph if is a complete graph on vertices. and interval bigraph when . We study the ordering characterization and forbidden obstructions of interval -graphs and present a polynomial-time recognition algorithm for them. Additionally, we discuss how this algorithm can be extended to recognize general interval -graphs. Special cases of interval -graphs, particularly comparability interval -graphs, were previously studied in [2], where the complexity interval -graph recognition was posed as an open problem.

Paper Structure

This paper contains 9 sections, 27 theorems, 1 equation, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction and Problem Definition
  2. Obstruction for interval $k$-graphs:
  3. Our Results and Future Work
  4. Basic definitions and some preliminary results
  5. Structural Properties of Strong components in $G^+$
  6. Interval $k$-graph recognition
  7. High level overview of the algorithm
  8. Proof of the correctness of the Algorithm
  9. Generalization

Key Result

Theorem 2

Let $G$ be a graph with a given partition of its vertices into $k$ partite sets $V_0, V_1, \dots,\newline V_{k-1}$. Then, it can be determined in $O(|V(G)||E(G)|)$ time whether $G$ is an interval $k$-graph.

Figures (8)

  • Figure 1: Forbidden patterns for interval bigraphs
  • Figure 2: Forbidden Patterns for co-circular arc bigraphs
  • Figure 3: Forbidden Patterns for bipartite permutation graphs (proper interval bigraphs)
  • Figure 4: Forbidden patterns for interval $k$-graphs.
  • Figure 5: Exobicliques: Here, $B=\{4,5,6,d,e,f\}$, $W=\{1,2,3,a,b,c\}$ and $B_1=\{d,e,f\}$, $W_1=\{1,2,3\}$ and $B \setminus B_1=\{4,5,6\}$, $W \setminus W_1=\{a,b,c\}$.
  • ...and 3 more figures

Theorems & Definitions (59)

  • Definition 1: interval $k$-graphs
  • Theorem 2
  • Lemma 5
  • proof
  • Lemma 6
  • Definition 7: circuit
  • Lemma 8
  • proof
  • Theorem 9
  • Lemma 10
  • ...and 49 more