Interval H-graphs : Recognition and forbidden obstructions

2025-03-02

AI Summary

This paper addresses the recognition problem for interval $H$-graphs: given a graph $G$, a $k$-partition of its vertices, and a target graph $H$, determine whether $G$ admits an interval representation on the real line such that two vertices are adjacent iff their parts are adjacent in $H$ and their corresponding intervals intersect. We establish the first order-theoretic characterization and a complete forbidden induced subgraph characterization for interval $k$-graphs, resolving the long-standing open question of their polynomial-time recognizability. We present the first $O(n^3)$-time recognition algorithm, combining greedy vertex ordering with nested interval validation. Furthermore, we generalize this framework to arbitrary interval $H$-graphs, yielding a scalable, structured recognition methodology. Our core contributions are threefold: (i) a complete theoretical characterization, (ii) an asymptotically optimal recognition algorithm, and (iii) a broadly applicable methodological framework for interval $H$-graph recognition.

Technology Category

• Planning, Routing, and Scheduling: Activity and Plan Recognition
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• Generalize interval graphs to interval H-graphs.
• Develop polynomial-time recognition for interval k-graphs.
• Identify forbidden obstructions in interval k-graphs.

Innovation

• Introduces interval H-graphs generalization
• Develops polynomial-time recognition algorithm
• Extends algorithm to general H-graphs