The Parameterized Complexity of Independent Set and More when Excluding a Half-Graph, Co-Matching, or Matching

A theorem of Ding, Oporowski, Oxley, and Vertigan implies that any sufficiently large twin-free graph contains a large matching, a co-matching, or a half-graph as a semi-induced subgraph. The sizes of these unavoidable patterns are measured by the matching index, co-matching index, and half-graph index of a graph. Consequently, graph classes can be organized into the eight classes determined by which of the three indices are bounded. We completely classify the parameterized complexity of Independent Set, Clique, and Dominating Set across all eight of these classes. For this purpose, we first derive multiple tractability and hardness results from the existing literature, and then proceed to fill the identified gaps. Among our novel results, we show that Independent Set is fixed-parameter tractable on every graph class where the half-graph and co-matching indices are simultaneously bounded. Conversely, we construct a graph class with bounded half-graph index (but unbounded co-matching index), for which the problem is W[1]-hard. For the W[1]-hard cases of our classification, we review the state of approximation algorithms. Here, we contribute an approximation algorithm for Independent Set on classes of bounded half-graph index.

💡 Research Summary

The paper investigates the parameterized complexity of three classic graph problems—Independent Set, Clique, and Dominating Set—under structural restrictions defined by three fundamental bipartite patterns: matchings, co‑matchings, and half‑graphs. For a graph G, the matching index, co‑matching index, and half‑graph index are the largest orders of semi‑induced subgraphs that are respectively a matching M_t, a co‑matching C_t, or a half‑graph H_t. A theorem of Ding, Oporowski, Oxley, and Vertigan (also proved by Alekseev and Gravier et al.) guarantees that any sufficiently large twin‑free graph must contain a large instance of at least one of these patterns, which motivates classifying graph families according to which of the three indices are bounded. This yields eight possible classes (each index either bounded or unbounded).

The authors provide a complete classification of the parameterized complexity of the three problems across all eight classes, summarized in Table 1. Known results cover many entries: when all three indices are bounded, the graph class has bounded neighborhood diversity, and all three problems are polynomial‑time solvable. When the matching index is bounded, the graph has bounded mim‑width, giving polynomial‑time algorithms for Independent Set and Dominating Set and trivial polynomial time for Clique. Fabiański et al. showed that Dominating Set is FPT on classes that are simultaneously half‑graph‑free and co‑matching‑free.

The main new contributions are:

1. Tractability – Theorem 4 proves that Independent Set is fixed‑parameter tractable on any class where both the half‑graph index and the co‑matching index are bounded (the so‑called semi‑ladder‑free or “half‑graph‑free ∧ co‑matching‑free” classes). The algorithm is based on indiscernibility techniques: long vertex sequences are identified such that external vertices are either adjacent to almost all or almost none of the sequence, enabling powerful reduction rules that shrink the instance while preserving the parameter.
2. Hardness – Theorem 2 constructs a graph class with half‑graph index at most 256 on which Independent Set remains W