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Conformal Hypergraphs: Duality and Implications for the Upper Clique Transversal Problem

Endre Boros, Vladimir Gurvich, Martin Milanič, Yushi Uno

TL;DR

This work investigates when the dual of a hypergraph is conformal (dually conformal hypergraphs) and provides complexity and algorithmic insights. It shows Dual Conformality lies in co-NP and gives polynomial-time recognition for bounded-dimension hypergraphs, plus a 2-SAT-based approach for the dimension-3 case; it also links these results to graph parameters via clique transversals, culminating in a polynomial-time algorithm for fixed $k$-Upper Clique Transversal. The paper further establishes a dimension-bounded characterization: a dually conformal Sperner hypergraph with dimension at most $k$ corresponds to a graph with upper clique transversal number at most $k$, enabling dimension-k algorithms with time $\\mathcal{O}(|E||V|^{2}\\Delta^{2}+|E||V|^{k})$. These results yield practical polynomial-time procedures for recognizing graphs with small maximal clique transversal sizes and raise open questions about the precise complexity of Dual Conformality and related graph classes.

Abstract

Given a hypergraph $\mathcal{H}$, the dual hypergraph of $\mathcal{H}$ is the hypergraph of all minimal transversals of $\mathcal{H}$. The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, etc. In this paper we study conformality of dual hypergraphs and prove several results related to the problem of recognizing this property. In particular, we show that the problem is in co-NP and can be solved in polynomial time for hypergraphs of bounded dimension. In the special case of dimension $3$, we reduce the problem to $2$-Satisfiability. Our approach has an implication in algorithmic graph theory: we obtain a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most $k$, for any fixed $k$.

Conformal Hypergraphs: Duality and Implications for the Upper Clique Transversal Problem

TL;DR

This work investigates when the dual of a hypergraph is conformal (dually conformal hypergraphs) and provides complexity and algorithmic insights. It shows Dual Conformality lies in co-NP and gives polynomial-time recognition for bounded-dimension hypergraphs, plus a 2-SAT-based approach for the dimension-3 case; it also links these results to graph parameters via clique transversals, culminating in a polynomial-time algorithm for fixed -Upper Clique Transversal. The paper further establishes a dimension-bounded characterization: a dually conformal Sperner hypergraph with dimension at most corresponds to a graph with upper clique transversal number at most , enabling dimension-k algorithms with time . These results yield practical polynomial-time procedures for recognizing graphs with small maximal clique transversal sizes and raise open questions about the precise complexity of Dual Conformality and related graph classes.

Abstract

Given a hypergraph , the dual hypergraph of is the hypergraph of all minimal transversals of . The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, etc. In this paper we study conformality of dual hypergraphs and prove several results related to the problem of recognizing this property. In particular, we show that the problem is in co-NP and can be solved in polynomial time for hypergraphs of bounded dimension. In the special case of dimension , we reduce the problem to -Satisfiability. Our approach has an implication in algorithmic graph theory: we obtain a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most , for any fixed .
Paper Structure (17 sections, 32 theorems, 6 equations, 2 figures, 1 table)

This paper contains 17 sections, 32 theorems, 6 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. Representation of hypergraphs
  5. Subtransverals
  6. Conformal hypergraphs
  7. Dually conformal hypergraphs
  8. Basic observations
  9. Computing the co-occurrence graph of the dual hypergraph
  10. The Dual Conformality problem
  11. A polynomial case of Dual Conformality
  12. Graphs with small upper clique transversal number
  13. Dually conformal hypergraphs with bounded dimension
  14. The general case
  15. The case of dimension three
  16. ...and 2 more sections

Key Result

Proposition 2.2

Given a hypergraph $\mathcal{H} = (V,E)$, with $V = \{v_1,\ldots, v_n\}$ and $E = \{e_1,\ldots, e_m\}$, there is an algorithm running in time $\mathcal{O}(|V||E|)$ that computes its edge-vertex incidence matrix and the doubly linked representation of its incident pairs. Using the incidence matrix w

Figures (2)

  • Figure 1: A hypergraph $\mathcal{H}$, its edge-vertex incidence matrix, and the doubly linked representation of its incident pairs.
  • Figure 2: A hypergraph $\mathcal{H}$, its dual hypergraph $\mathcal{H}^d$, and the co-occurrence graph of $\mathcal{H}^d$.

Theorems & Definitions (61)

  • Proposition 2.2
  • Theorem 2.3: Boros, Gurvich, Elbassioni, and Khachiyan MR1841121
  • Corollary 2.4
  • proof
  • Theorem 2.6: BeeriFMY83; see also zbMATH03400923MR1013569
  • Lemma 2.7
  • proof
  • Theorem 2.8
  • proof
  • Theorem 2.9: Gilmore gilmore1962families; see also zbMATH03485854zbMATH03400923MR1013569
  • ...and 51 more