Parameterized Complexity of Dominating Set Variants in Almost Cluster and Split Graphs

TL;DR

The paper investigates how the Dominating Set problem and its variants behave under deletion-distance parameters to simple graph classes, focusing on distance to cluster graphs (CVD) and split graphs (SVD). It develops unified FPT algorithms via a Set-Cover with Partition framework, achieving running times of 3^k n^{O(1)} for DS, IDS, DC, EDS, and TDS when parameterized by CVD, and (r+2)^k n^{O(1)} for ThDS; for SVD, IDS and EDS admit 2^{|S|} n^{O(1)} and 3^{k/2} n^{O(1)} respectively. The work also presents tight lower bounds under ETH/SETH and kernelization results, showing no polynomial kernels for several cluster-distance variants and establishing strong hardness for IDS/EDS under these structural parameters. Collectively, the results advance the understanding of how structural graph parameters influence the tractability of classic domination problems and introduce techniques that may generalize to other Set-Cover-like problems. The findings have implications for both theory and practical algorithm design in scenarios where inputs are close to cluster or split structures.

Abstract

We consider structural parameterizations of the fundamental Dominating Set problem and its variants in the parameter ecology program. We give improved FPT algorithms and lower bounds under well-known conjectures for dominating set in graphs that are k vertices away from a cluster graph or a split graph. These are graphs in which there is a set of k vertices (called the modulator) whose deletion results in a cluster graph or a split graph. We also call k as the deletion distance (to the appropriate class of graphs). When parameterized by the deletion distance k to cluster graphs - we can find a minimum dominating set (DS) in 3^k n^{O(1)}-time. Within the same time, we can also find a minimum independent dominating set (IDS) or a minimum dominating clique (DC) or a minimum efficient dominating set (EDS) or a minimum total dominating set (TDS). We also show that most of these variants of dominating set do not have polynomial sized kernel. Additionally, we show that when parameterized by the deletion distance k to split graphs - IDS can be solved in 2^k n^{O(1)}-time and EDS can be solved in 3^{k/2}n^{O(1)}.

Figures (5)

• Figure 1: An illustration of the clause gadget described in Section \ref{['thm:lower-bound-eds-vc-new']}. The vertices are $c_1,c_2,c_3,d_0,d_1,d_2,d_3,d_{1,2},d_{2,3},d_{1,3}$. The edges are as follows: (i) $c_1$ is adjacent to $d_1, d_{1,2}$ and $d_{1,3}$, (ii) $c_2$ is adjacent to $d_2, d_{1,2}$ and $d_{2,3}$, (iii) make $c_3$ adjacent to $d_3, d_{2, 3}$ and $d_{1,3}$, (iv) make $d_0$ adjacent to each of the six vertices $d_1, d_2, d_3, d_{1,2}, d_{2,3}$, and $d_{1,3}$, and (v) $\{d_0, d_1, d_2, d_3, d_{1,2}, d_{2,3}, d_{1,3}\}$ is a clique.
• Figure 2: Illustration of Branching Rule \ref{['branch-rule:branch-on-pair-from-S']}. Note that the number of blue vertices drops by at least two in each of the branches.
• Figure 3: Illustration of Branching Rule \ref{['branch-rule:pair-of-non-adjacent']} for the first case. Note that the number of blue vertices in $S$ drops by at least two in each of the branches.
• Figure 4: Illustration of Branching Rule \ref{['branch-rule:pair-of-non-adjacent']} for the second case. Note that the number of blue vertices in $S$ drops by two in each of the branches.
• Figure 5: An illustration of Lemma \ref{['lem:equal-neighbor-hood-property']}. The red vertices are $N(v) \setminus \{u\} \subseteq S$ and $N(u) \setminus \{v\} \subseteq S$.

Theorems & Definitions (41)

• Proposition 2.1: bodlaender2011kernel
• Conjecture 2.1: Strong Exponential Time Hypothesis (SETH) IPZ01
• Conjecture 2.2: Exponential Time Hypothesis (ETH) IPZ01IP01
• Conjecture 2.3: Set Cover Conjecture (SCC) cygan2016problems
• proof
• proof
• proof
• Corollary 3.1
• proof
• proof