Antony, D and Chandran, LS and Gayen, A and Gosavi, S and Jacob, S
(2024)
*Total Domination, Separated-Cluster, CD-Coloring: Algorithms andÂ Hardness.*
In: 16th Latin American Symposium on Theoretical Informatics, LATIN 2042, 18 March 2024through 22 March 2024, Puerto Varas, pp. 97-113.

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## Abstract

Domination and coloring are two classic problems in graph theory. In this paper, our major focus is on the CD-coloring problem, which incorporates the flavors of both domination and coloring in it. Let G be an undirected graph. A proper vertex coloring of G is said to be a cd-coloring, if each color class has a dominating vertex in G. The minimum integer k for which there exists a cd-coloring of G using k colors is called the cd-chromatic number of G, denoted as Ï�cd(G). A set Sâ��V(G) is said to be a total dominating set, if any vertex in G has a neighbor in S. The total domination number of G, denoted as Î³t(G), is defined to be the minimum integer k such that G has a total dominating set of size k. A set Sâ��V(G) is said to be a separated-cluster (also known as sub-clique) if no two vertices in S lie at a distance exactly 2 in G. The separated-cluster number of G, denoted as Ï�s(G), is defined to be the maximum integer k such that G has a separated-cluster of size k. In this paper, we contribute to the literature connecting CD-coloring with the problems, Total Domination and Separated-Cluster. For any graph G, we have Ï�cd(G)â�¥Î³t(G) and Ï�cd(G)â�¥Ï�s(G). First, we explore the connection of CD-Coloring problem to the well-known problem Total Domination. Note that Total Domination is known to be NP-Complete for triangle-free 3-regular graphs. We generalize this result by proving that both the problems CD-Coloring and Total Domination are NP-Complete, and do not admit any subexponential-time algorithms on triangle-freed-regular graphs, for each fixed integerdâ�¥3, assuming the Exponential Time Hypothesis. We also study the relationship between the parameters Ï�cd(G) and Ï�s(G). Analogous to the well-known notion of â��perfectnessâ��, here we introduce the notion of â��cd-perfectnessâ��. We prove a sufficient condition for a graph G to be cd-perfect (i.e. Ï�cd(H)=Ï�s(H), for any induced subgraph H of G). Our sufficient condition is also necessary for certain graph classes (like triangle-free graphs). This unified approach of â��cd-perfectnessâ�� has several exciting consequences. In particular, it is interesting to note that the same framework can be used as a tool to derive both positive and negative results concerning the algorithmic complexity of CD-coloring and Separated-Cluster. Â© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

Item Type: | Conference Paper |
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Publication: | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher: | Springer Science and Business Media Deutschland GmbH |

Additional Information: | The copyright for this article belongs to Springer Science and Business Media Deutschland GmbH. |

Keywords: | Color; Graphic methods; Undirected graphs, CD-coloring; CD-perfectness; Coloring problems; Graph G; Regular graphs; Separated-cluster; Total dominating sets; Total domination; Triangle-free; Triangle-free d-regular graph, Coloring |

Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |

Date Deposited: | 30 Aug 2024 12:09 |

Last Modified: | 30 Aug 2024 12:09 |

URI: | http://eprints.iisc.ac.in/id/eprint/84925 |

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