Total Domination, Separated Clusters, CD-Coloring: Algorithms and Hardness

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, \(\chi_{cd}(G)\). A set \(S\subseteq V(G)\) is a total dominating set if any vertex in \(G\) has a neighbor in \(S\). The total domination number, \(\gamma_t(G)\) of \(G\) is the minimum integer \(k\) such that \(G\) has a total dominating set of size \(k\). A set \(S\subseteq V(G)\) is a \(separated-cluster\) if no two vertices in \(S\) lie at a distance 2 in \(G\). The separated-cluster number, \(\omega_s(G)\), 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 \(d\geq 3\). We also study the relationship between the parameters \(\chi_{cd}(G)\) and \(\omega_s(G)\). 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. \(\chi_{cd}(H)= \omega_s(H)\), 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.