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We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), 330-338) by proving that it is NP-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is always possible for every maximal outerplanar graph with at least three vertices. Moreover, we extend their previous result by proving that deciding whether a bipartite graph can be partitioned into k independent dominating sets is NP-complete for every k ≥ 3. We also strengthen a result by Henning et al. (Discrete Math. (2009), 6451-6458) by showing that it is NP-complete to determine if a graph has two disjoint independent dominating sets, even when the problem is restricted to triangle-free planar graphs. Finally, for every k ≥ 3, we show that there is some constant t depending only on k such that deciding whether a k-regular graph can be partitioned into t independent dominating sets is NP-complete. We conclude by deriving moderately exponential-time algorithms for the problem.

A perfect Italian dominating function of a graph G = (V, E) is a function f : V → {0, 1, 2} such that for every vertex f (v) = 0, it holds that u∈N (v) f (u) = 2, i.e., the weight of the labels assigned by f to the neighbors of v is exactly two. The weight of a perfect Italian function is the sum of the weights of the vertices. The perfect Italian domination number of G, denoted by γ p I (G), is the minimum weight of any perfect Italian dominating function of G. While introducing the parameter, Haynes and Henning (Discrete Appl. Math. ( 2019), 164-177) also proposed the problem of determining the best possible constants c G such that γ p I (G) ≤ c G × n for all graphs of order n when G is in a particular class G of graphs. They proved that c G = 1 when G is the class of bipartite graphs, and raised the question for planar graphs and regular graphs. We settle their question precisely for planar graphs by proving that c G = 1 and for cubic graphs by proving that c G = 2/3. For split graphs, we also show that c G = 1. In addition, we characterize the graphs G with γ p I (G) equal to 2 and 3 and determine the exact value of the parameter for several simple structured graphs. We conclude by proving that it is NP-complete to decide whether a given bipartite planar graph admits a perfect Italian dominating function of weight k.

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An exact (k, d)-coloring of a graph G is a coloring of its vertices with k colors such that each vertex v is adjacent to exactly d vertices having the same color as v. The exact d-defective chromatic number, denoted χ = d (G), is the minimum k such that there exists an exact (k, d)-coloring of G. In an exact (k, d)-coloring, which for d = 0 corresponds to a proper coloring, each color class induces a d-regular subgraph. We give basic properties for the parameter and determine its exact value for cycles, trees, and complete graphs. In addition, we establish bounds on χ = d (G) for all relevant values of d when G is planar, chordal, or has bounded treewidth. We also give polynomial-time algorithms for finding certain types of exact (k, d)-colorings in cactus graphs and block graphs. Our main result is on the computational complexity of d-Exact Defective k-Coloring in which we are given a graph G and asked to decide whether χ = d (G) ≤ k. Specifically, we prove that the problem is NP-complete for all d ≥ 1 and k ≥ 2.