Interval H-graphs : Recognition and forbidden obstructions
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866909520293265408 |
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| author | Müller, Haiko Rafiey, Arash |
| author_facet | Müller, Haiko Rafiey, Arash |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00672 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Interval H-graphs : Recognition and forbidden obstructions Müller, Haiko Rafiey, Arash Discrete Mathematics Data Structures and Algorithms Combinatorics 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. |
| title | Interval H-graphs : Recognition and forbidden obstructions |
| topic | Discrete Mathematics Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2503.00672 |