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

TL;DR

This work studies the interplay between cd-coloring, total domination, and separated-cluster in graphs by introducing cd-perfectness and leveraging an auxiliary graph $G^*$ to relate $\chi_{cd}(G)$ and $\omega_s(G)$. It establishes NP-completeness of CD-coloring on triangle-free $d$-regular graphs for all fixed $d\ge 3$, implying NP-hardness for Total Domination in the same class and providing ETH-based lower bounds. A cd-perfectness framework yields polynomial-time results for several graph classes (e.g., chordal bipartite graphs) and identifies hardness in others (e.g., $C_6$-free bipartite graphs), offering a unified approach to algorithmic questions. Additionally, Separation-Cluster is shown to be polynomial on interval graphs by reducing to max weighted independent set on cocomparability graphs, solving an open problem and highlighting the role of $G^*$ in bridging these problems. The paper thus advances understanding of when domination-coloring variants coincide and how structure (via $G^*$ and cd-perfectness) drives tractability across graph families.

Abstract

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 $S\subseteq V(G)$ 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 $S\subseteq V(G)$ 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 $d\geq 3$. 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.

Figures (10)

• Figure 1: Set of graphs in $\mathcal{H}$
• Figure 2: The gadget $W$ used in Construction \ref{['cons:cubic']}. The shaded vertices, $d,b,f$ and $c$, respectively dominates the color classes, $\{b,h,i\},\{e,a,d\},\{c,j,k\}$, and $\{g,f\}$, in a valid $cd$-coloring of $W$.
• Figure 3: An example of a bipartite graph $G$ with bounded degree 3
• Figure 4: An example of the resultant graph $G_c$ obtained from Construction \ref{['cons:cubic']} corresponding to the bipartite graph $G$ with bounded degree 3 in Figure \ref{['fig:bipartite']}. The dotted rectangles represent the gadget $W$ used for the construction.
• Figure 5: An example of the gadget $W$ used in Construction \ref{['cons:d-regular odd']}, for $d=5$ such that the shaded vertices correspond to the dominating vertices of the color classes in a valid $cd$-coloring of $W$.

Theorems & Definitions (58)

• Theorem 1
• Corollary 2
• Definition 1: Auxiliary graph $G^*$ ShaluSandhyaLowBound17
• Theorem 3
• proof
• Proposition 9: ZhuApproxMinTotDom09
• Proposition 10: MerouaneHCK15
• proof
• Lemma 12
• proof