Total Domination, Separated Clusters, CD-Coloring: Algorithms and Hardness
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Domination and coloring are two classic problems in graph theory. The major focus of this paper is the CD-COLORING problem which combines the flavours of domination and colouring. Let G be an undirected graph. A proper vertex coloring of G is 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, χ_cd(G). A set S⊆ V(G) is a total dominating set if any vertex in G has a neighbor in S. The total domination number, γ_t(G) of G is the minimum integer k such that G has a total dominating set of size k. A set S⊆ V(G) is a separated-cluster if no two vertices in S lie at a distance 2 in G. The separated-cluster number, ω_s(G), of G is the maximum integer k such that G has a separated-cluster of size k. In this paper, first we explore the connection between CD-COLORING and TOTAL DOMINATION. We prove that CD-COLORING and TOTAL DOMINATION are NP-Complete on triangle-free d-regular graphs for each fixed integer d≥ 3. We also study the relationship between the parameters χ_cd(G) and ω_s(G). Analogous to the well-known notion of `perfectness', here we introduce the notion of `cd-perfectness'. We prove a sufficient condition for a graph G to be cd-perfect (i.e. χ_cd(H)= ω_s(H), for any induced subgraph H of G) which is also necessary for certain graph classes (like triangle-free graphs). Here, we propose a generalized framework via which we obtain several exciting consequences in the algorithmic complexities of special graph classes. In addition, we settle an open problem by showing that the SEPARATED-CLUSTER is polynomially solvable for interval graphs.
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