Parameterized algorithms for conflict-free colorings of graphs

Abstract

In this paper, we study the conflict-free coloring of graphs induced by neighborhoods. A coloring of a graph is conflict-free if every vertex has a uniquely colored vertex in its neighborhood. The conflict-free coloring problem is to color the vertices of a graph using the minimum number of colors such that the coloring is conflict-free. We consider both closed neighborhoods, where the neighborhood of a vertex includes itself, and open neighborhoods, where a vertex does not included in its neighborhood. We study the parameterized complexity of conflict-free closed neighborhood coloring and conflict-free open neighborhood coloring problems. We show that both problems are fixed-parameter tractable (FPT) when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Gargano et al. (2015) that conflict-free coloring is fixed-parameter tractable parameterized by the vertex cover number. Also, we show that both problems admit an additive constant approximation algorithm when parameterized by the distance to threshold graphs. We also study the complexity of the problem on special graph classes. We show that both problems can be solved in polynomial time on cographs. For split graphs, we give a polynomial time algorithm for closed neighborhood conflict-free coloring problem, whereas we show that open neighborhood conflict-free coloring is NP-complete. We show that interval graphs can be conflict-free colored using at most four colors.

Combinatorial Bounds for Conflict-free Coloring on Open Neighborhoods

“…4. In Section 5.2, we show that χ ON (G) ≤ dc(G)+3, where dc(G) is the distance to cluster parameter of G. This is an improvement over the previous bound [5] of 2dc(G) + 3. Our bound is nearly tight since there are graphs for which χ ON (G) = dc(G).…”

“…For the results in terms of parameters neighborhood diversity and distance to cluster, the obvious open questions are to improve the bounds and/or to provide tight examples. 5. When G is planar, we show that χ * ON (G) ≤ 5.…”

“…Since the problem is NP-hard, the parameterized aspects of the problem have been studied. The problems are fixed parameter tractable when parameterized by vertex cover number, neighborhood diversity [4], distance to cluster [5], and more recently, treewidth [6,7]. This problem has attracted special interest for graphs arising out of intersection of geometric objects, see for instance, [8,9,10].…”

Conflict-Free Coloring on Open Neighborhoods

“…Since the problem is NP-hard, the parameterized aspects of the problem have been studied. The problems are fixed parameter tractable when parameterized by vertex cover number, neighborhood diversity [4], distance to cluster, distance to threshold graphs [5], and more recently, tree-width [6,7].…”

“…Of these, graphs arising out of intersection of geometric objects have attracted special interest, see for instance, [9,11,12]. The problem has also been studied for structural classes of graphs such as bipartite graphs and split graphs [5].…”