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Clique-free t-matchings in degree-bounded graphs

Katarzyna Paluch, Mateusz Wasylkiewicz

TL;DR

This work addresses finding maximum size or maximum weight $t$-matchings in graphs of maximum degree $t+1$ that avoid forbidden complete partite subgraphs, notably $K^p_q$, $K_{t+1}$, and $K_{t,t}$. The authors introduce a complement-based reduction that replaces forbidden subgraphs with gadgets containing half-edges, mapping restricted or $K^p_q$-free $t$-matchings in the original graph to suitable $(l,b)$-matchings in an augmented graph $G'$, and then reconstruct the desired solution from the complement. They give simple, fast combinatorial algorithms for both weighted (edge weights vertex-induced on forbidden subgraphs) and unweighted versions, achieving $O(\min\{nm\log n, n^3\})$ time in the weighted case and $O(\sqrt{n}m)$ for unweighted, with extensions to a new $k$-restricted $t$-matching variant. The methods unify and generalize prior work on triangle-free and square-free $t$-matchings, offering a versatile toolkit (gadgets, half-edges, potentials) for broader forbidden-subgraph constraints and potential applications to connectivity augmentation problems. The results advance the tractability frontier for weighted constrained $t$-matching problems in degree-bounded graphs and provide practical, scalable algorithms for related combinatorial optimization tasks.

Abstract

We consider problems of finding a maximum size/weight $t$-matching without forbidden subgraphs in an undirected graph $G$ with the maximum degree bounded by $t+1$, where $t$ is an integer greater than $2$. Depending on the variant forbidden subgraphs denote certain subsets of $t$-regular complete partite subgraphs of $G$. A graph is complete partite if there exists a partition of its vertex set such that every pair of vertices from different sets is connected by an edge and vertices from the same set form an independent set. A clique $K_t$ and a bipartite clique $K_{t,t}$ are examples of complete partite graphs. These problems are natural generalizations of the triangle-free and square-free $2$-matching problems in subcubic graphs. In the weighted setting we assume that the weights of edges of $G$ are vertex-induced on every forbidden subgraph. We present simple and fast combinatorial algorithms for these problems. The presented algorithms are the first ones for the weighted versions, and for the unweighted ones, are faster than those known previously. Our approach relies on the use of gadgets with so-called half-edges. A half-edge of edge $e$ is, informally speaking, a half of $e$ containing exactly one of its endpoints.

Clique-free t-matchings in degree-bounded graphs

TL;DR

This work addresses finding maximum size or maximum weight -matchings in graphs of maximum degree that avoid forbidden complete partite subgraphs, notably , , and . The authors introduce a complement-based reduction that replaces forbidden subgraphs with gadgets containing half-edges, mapping restricted or -free -matchings in the original graph to suitable -matchings in an augmented graph , and then reconstruct the desired solution from the complement. They give simple, fast combinatorial algorithms for both weighted (edge weights vertex-induced on forbidden subgraphs) and unweighted versions, achieving time in the weighted case and for unweighted, with extensions to a new -restricted -matching variant. The methods unify and generalize prior work on triangle-free and square-free -matchings, offering a versatile toolkit (gadgets, half-edges, potentials) for broader forbidden-subgraph constraints and potential applications to connectivity augmentation problems. The results advance the tractability frontier for weighted constrained -matching problems in degree-bounded graphs and provide practical, scalable algorithms for related combinatorial optimization tasks.

Abstract

We consider problems of finding a maximum size/weight -matching without forbidden subgraphs in an undirected graph with the maximum degree bounded by , where is an integer greater than . Depending on the variant forbidden subgraphs denote certain subsets of -regular complete partite subgraphs of . A graph is complete partite if there exists a partition of its vertex set such that every pair of vertices from different sets is connected by an edge and vertices from the same set form an independent set. A clique and a bipartite clique are examples of complete partite graphs. These problems are natural generalizations of the triangle-free and square-free -matching problems in subcubic graphs. In the weighted setting we assume that the weights of edges of are vertex-induced on every forbidden subgraph. We present simple and fast combinatorial algorithms for these problems. The presented algorithms are the first ones for the weighted versions, and for the unweighted ones, are faster than those known previously. Our approach relies on the use of gadgets with so-called half-edges. A half-edge of edge is, informally speaking, a half of containing exactly one of its endpoints.
Paper Structure (17 sections, 24 theorems, 4 equations, 4 figures, 2 algorithms)

This paper contains 17 sections, 24 theorems, 4 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our results
  3. Related work
  4. Preliminaries
  5. Outline of the Algorithm
  6. The bounded restricted $t$-matching problem
  7. Disjoint forbidden subgraphs
  8. Non-disjoint forbidden subgraphs
  9. The bounded $K^p_q$-free $t$-matching problem
  10. Case $q\geq 3$
  11. Case $q=2$
  12. Running time analysis
  13. Computing a minimum weight $(l,b)$-matching of $G'$ in the weighted case
  14. Computing a minimum weight $(l,b)$-matching of $G'$ in the unweighted case
  15. Finding all forbidden subgraphs of $G$
  16. ...and 2 more sections

Key Result

Theorem 1

There is an algorithm that, given a multigraph $G=(V,E)$ (i.e. $G$ may contain parallel edges), a weight function $w:E\to\mathbb{R}$ and vectors $l,b\in\mathbb{N}^V$, finds a maximum weight $(l,b)$-matching of $G$ in time $O\left((\sum_{v\in V}b(v))\min\{|E|\log{|V|},|V|^2\}\right)$, assuming that $

Figures (4)

  • Figure 1: A gadget for $K_{t+1}$ for $t=5$.
  • Figure 2: A gadget for $K_{t,t}$ for $t=3$.
  • Figure 3: A gadget for $K^p_q$ for $p=3$ and $q=3$.
  • Figure 4: A gadget for a dense subgraph consisting of eight vertices with the core of size four whose center $c_H$ has nonnegative potential.

Theorems & Definitions (64)

  • Theorem 1: Gabow1983
  • Claim 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 54 more