Partial Domination in Some Geometric Intersection Graphs and Some Complexity Results

TL;DR

This work studies the partial domination problem, a generalization of the classic dominating set, and the maximum dominating k-set, establishing their computational equivalence. It proves NP-hardness for α-partial domination on general graphs while identifying graph classes where domination remains easy but partial domination is hard. The authors then develop polynomial-time algorithms for maximum dominating k-set on several geometric intersection graphs, including unit interval and general interval graphs, as well as special cases of unit squares, axis-parallel unit-height rectangles, and disks intersected by a line, with runtimes that depend on structural parameters like box counts and diameter ratios. The results advance understanding of tractable partial domination in restricted geometric settings and propose parameterized and strip-based DP techniques that could extend to additional object families. These findings have potential implications for network coverage with limited resources and related combinatorial optimization problems on geometric graphs.

Abstract

{\em Partial domination problem} is a generalization of the {\em minimum dominating set problem} on graphs. Here, instead of dominating all the nodes, one asks to dominate at least a fraction of the nodes of the given graph by choosing a minimum number of nodes. For any real number $α\in(0,1]$, $α$-partial domination problem can be proved to be NP-complete for general graphs. In this paper, we define the {\em maximum dominating $k$-set} of a graph, which is polynomially transformable to the partial domination problem. The existence of a graph class for which the minimum dominating set problem is polynomial-time solvable, whereas the partial dominating set problem is NP-hard, is shown. We also propose polynomial-time algorithms for the maximum dominating $k$-set problem for the unit and arbitrary interval graphs. The problem can also be solved in polynomial time for the intersection graphs of a set of 2D objects intersected by a straight line, where each object is an axis-parallel unit square, as well as in the case where each object is a unit disk. Our technique also works for axis-parallel unit-height rectangle intersection graphs, where a straight line intersects all the rectangles. Finally, a parametrized algorithm for the maximum dominating $k$-set problem in a disk graph where the input disks are intersected by a straight line is proposed; here the parameter is the ratio of the diameters of the largest and smallest input disks.

Figures (16)

• Figure 1: A given $2$-CNF expression $\cal C$ and its corresponding graph $G_{\cal C}=(V_{\cal C}, E_{\cal C})$
• Figure 2: An interval layout and its corresponding interval graph
• Figure 3: A unit interval graph consisting of $6$ vertices
• Figure 4: Here, $[a_1,b_1],[a_2,b_2],[a_3,b_3]$ and $[a_4,b_4]$ are non-deleted intervals, and the rest are marked as deleted intervals. All these intervals lie on the real line (dashed line), but they have been drawn in the two-dimensional plane for visual clarity.
• Figure 5: Here, the width of the required strip is $2AC$.

Theorems & Definitions (24)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Theorem 2
• Corollary 1
• Lemma 1
• Lemma 2
• Theorem 3
• Theorem 4