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

TL;DR

The paper addresses the parameterized complexity of Independent Set, Clique, and Dominating Set under forbiddance of semi-induced patterns (matching, co-matching, half-graph) and introduces three indices to capture these patterns. It provides a complete eight-class classification, including a key tractability result: Independent Set is fixed-parameter tractable when both the half-graph and co-matching indices are bounded, alongside a constructed half-graph-free class where IS is W[1]-hard and approximation results for IS on restricted classes. The approach combines indiscernible-sequence kernelization, reductions, and structural lemmas that connect model-theoretic stability concepts with algorithmic tractability, as well as reductions that separate tractability boundaries. Overall, the work unifies sparsity, width-measures, and model-theoretic perspectives to map precise tractability and hardness boundaries for these classical problems on structurally restricted graph classes, while offering practical approximation techniques in restricted settings.

Abstract

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.

Figures (4)

• Figure 1: A hierarchy of graph class properties. An arrow $P_1 \to P_2$ means every graph class with property $P_1$ also has property $P_2$.
• Figure 2: A positive instance of Grid Tiling with $k = 3$ and $n=6$. The table displays the cells $\pi=(i,j)$ and lists the tiles contained in $S_\pi \subseteq [6]^2$. A feasible solution is marked in bold.
• Figure 3: We guess a vertex $v$ whose neighborhood contains a large part of an independent set and an adjacent vertex $u$ that is part of the independent set. We recurse into the graph $G_{uv} = G[N(v) \setminus N[u]]$ whose half-graph index has decreased.
• Figure 4: The connection patterns of the vertex $y$ (on the top) towards a tuple from the indiscernible sequence (on the bottom) that is verified by the formulas $\chi_{2t}, \chi_{2t}^\star, \theta_{2t},\delta_{2t}, \delta^\star_{2t}$ where $t = 4$.

Theorems & Definitions (39)

• Theorem 1: DBLP:journals/jct/DingOOV96alekseev1997lowergravier2004
• Corollary 1
• Theorem 2
• proof
• Claim 1
• proof : Proof
• Claim 2
• proof : Proof
• Theorem 3
• proof