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Domination in Diameter-Two Graphs and the 2-Club Cluster Vertex Deletion Parameter

Faisal N. Abu-Khzam, Lucas Isenmann

TL;DR

This work investigates domination problems on diameter-$2$ graphs through the lens of the $2ccvd$ parameter (the 2-club cluster vertex deletion number). It establishes an FPT algorithm for Efficient Dominating Set parameterized by $2ccvd$ and proves hardness results for Independent Dominating Set and several Roman domination variants on the same graph class, highlighting a nuanced boundary between tractability and intractability under diameter-two constraints. The authors also propose the Fixed-Parameter Sub-Exponential (FPSUB) framework, showing sub-exponential algorithms for certain problems (e.g., Dominating Set, 3-Coloring) and discussing ETH-based limitations for others, thereby outlining future directions for FPSUB in parameterized graph problems. Overall, the paper advances understanding of how restricting graph structure to diameter-two can both enable efficient algorithms for some domination problems and preserve hardness for others, while introducing FPSUB as a promising paradigm for sub-exponential fixed-parameter complexity research.

Abstract

The s-club cluster vertex deletion number of a graph, or sccvd, is the minimum number of vertices whose deletion results in a disjoint union of s-clubs, or graphs whose diameter is bounded above by s. We launch a study of several domination problems on diameter-two graphs, or 2-clubs, and study their parameterized complexity with respect to the 2ccvd number as main parameter. We further propose to explore the class of problems that become solvable in sub-exponential time when the running time is independent of some input parameter. Hardness of problems for this class depends on the Exponential-Time Hypothesis. We give examples of problems that are in the proposed class and problems that are hard for it.

Domination in Diameter-Two Graphs and the 2-Club Cluster Vertex Deletion Parameter

TL;DR

This work investigates domination problems on diameter- graphs through the lens of the parameter (the 2-club cluster vertex deletion number). It establishes an FPT algorithm for Efficient Dominating Set parameterized by and proves hardness results for Independent Dominating Set and several Roman domination variants on the same graph class, highlighting a nuanced boundary between tractability and intractability under diameter-two constraints. The authors also propose the Fixed-Parameter Sub-Exponential (FPSUB) framework, showing sub-exponential algorithms for certain problems (e.g., Dominating Set, 3-Coloring) and discussing ETH-based limitations for others, thereby outlining future directions for FPSUB in parameterized graph problems. Overall, the paper advances understanding of how restricting graph structure to diameter-two can both enable efficient algorithms for some domination problems and preserve hardness for others, while introducing FPSUB as a promising paradigm for sub-exponential fixed-parameter complexity research.

Abstract

The s-club cluster vertex deletion number of a graph, or sccvd, is the minimum number of vertices whose deletion results in a disjoint union of s-clubs, or graphs whose diameter is bounded above by s. We launch a study of several domination problems on diameter-two graphs, or 2-clubs, and study their parameterized complexity with respect to the 2ccvd number as main parameter. We further propose to explore the class of problems that become solvable in sub-exponential time when the running time is independent of some input parameter. Hardness of problems for this class depends on the Exponential-Time Hypothesis. We give examples of problems that are in the proposed class and problems that are hard for it.
Paper Structure (9 sections, 16 theorems, 3 equations, 3 figures, 3 tables, 2 algorithms)

This paper contains 9 sections, 16 theorems, 3 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Efficient Dominating Set parameterized by 2ccvd
  4. Complexity of Independent Dominating Set on $2$-clubs
  5. Roman domination
  6. Complexity of Roman domination in 2-Clubs
  7. Complexity of Independent Roman Domination in 2-clubs
  8. Complexity of Perfect Roman Domination in $2$-clubs
  9. Fixed-Parameter Sub-Exponential Algorithms

Key Result

Lemma 1

A subset $D$ of $V(G)$ is an EDS of $G$ such that $S' = D \cap S$ if and only if there exists $(v_1, \ldots, v_p) \in V(C_1) \times \cdots \times V(C_p)$ such that

Figures (3)

  • Figure 1: Example of a graph obtained by the construction from an instance of $k$- Multicolored Independent Set.
  • Figure 2: Example of the duplication construction: The depicted graph is the graph $G'$ obtained from a graph $G$ which is $3$-path with vertex set $\{u,v,w\}$.
  • Figure 3: Example of a graph obtained by the construction of Theorem \ref{['theorem:PRD']} for the Exact Cover instance $(U=\{u_1, u_2, u_3\}, S=\{s_1=\{u_1, u_2\}, s_2 =\{u_1, u_3\}, s_3=\{u_2, u_3\})$. For readability, independent sets $U_j$, $S_i$ and $T_i$ are represented by a single blue vertex and some edges incident to $a,b$ and $c$ are not described.

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • Theorem 2: Goyal et al. goyal2024parameterized
  • ...and 14 more