The Domination and Independent Domination Problems in Supergrid Graphs

Hung, Ruo-Wei; Chiu, Ming-Jung; Chen, Jong-Shin

doi:10.1007/978-3-030-86653-2_46

Cited by 3 publications

(4 citation statements)

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“…However, their complexities for diagonal supergrid, triangular supergrid, and supergrid graphs remain unknown. In [28], we have provided a rough proof to claim that they are NP-complete for supergrid graphs. In this section, we present a complete proof to show that they are NP-complete for diagonal supergrid graphs.…”

Section: Np-completeness Resultsmentioning

confidence: 99%

“…For instance, the arrow lines in Figure 3b depict a horizontal path P 9 . In [28,34], the domination number γ(R m×n ) and the independent domination number γ ind (R m×n ) of R m×n are computed as follows:…”

Section: Preliminariesmentioning

confidence: 99%

“…Thus, the studied problems are NPcomplete as well for extended supergrid graphs. In [28], we provided a rough proof to claim that the domination and independent domination problems on supergrid graphs are NP-complete. In this paper, we present a full proof to show that the domination and independent domination problems on diagonal supergrid graphs are NP-complete.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, the domination and independent domination numbers of rectangular grid graphs have been computed in [18,[29][30][31]. In [28], we solved the domination and independent domination problems on rectangular supergrid graphs in linear time. However, the complexities of the domination and independent domination problems on rectangular triangular-supergrid graphs remain unknown.…”

Section: Introductionmentioning

confidence: 99%

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Domination and Independent Domination in Extended Supergrid Graphs

et al. 2022

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Supergrid graphs are derived by computing stitch paths for computerized embroidery machines. In the past, we have studied the Hamiltonian-related properties of supergrid graphs and their subclasses of graphs. In this paper, we propose a generalized graph class for supergrid graphs called extended supergrid graphs. Extended supergrid graphs include grid graphs, supergrid graphs, diagonal supergrid graphs, and triangular supergrid graphs as subclasses of graphs. In this paper, we study the problems of domination and independent domination on extended supergrid graphs. A dominating set of a graph is the subset of vertices on it, such that every vertex of the graph is in this set or adjacent to at least a vertex of this set. If any two vertices in a dominating set are not adjacent, this is called an independent dominating set. Domination and independent domination problems find a dominating set and an independent dominating set with the least number of vertices on a graph, respectively. The domination and independent domination set problems on grid graphs are known to be NP-complete, meaning that these two problems on extended supergrid graphs are also NP-complete. However, the complexities of these two problems in other subclasses of graphs remain unknown. In this paper, we first prove that these two problems on diagonal supergrid graphs are NP-complete, then, by a simple extension, we prove that these two problems on supergrid graphs and triangular supergrid graphs are also NP-complete. In addition, these two problems on rectangular supergrid graphs are known to be linearly solvable; however, the complexities of these two problems on rectangular triangular-supergrid graphs remain unknown. This paper provides tight upper bounds on the sizes of the minimum dominating and independent dominating sets for rectangular triangular-supergrid graphs.

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Section: Np-completeness Resultsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%