Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196

Efficient domination and polarity

The thesis considers the following graph problems: Efficient (Edge) Domination seeks for an independent vertex (edge) subset D such that all other vertices (edges) have exactly one neighbor in D. Polarity asks for a vertex subset that induces a complete multipartite graph and that contains a vertex of every induced P_3. Monopolarity is the special case of Polarity where the wanted vertex subset has to be independent. These problems are NP-complete in general, but efficiently solvable on various graph classes. The thesis sharpens known NP-completeness results and presents new solvable cases

Tropical Dominating Sets in Vertex-Coloured Graphs

Given a vertex-coloured graph, a dominating set is said to be tropical if every colour of the graph appears at least once in the set. Here, we study minimum tropical dominating sets from structural and algorithmic points of view. First, we prove that the tropical dominating set problem is NP-complete even when restricted to a simple path. Then, we establish upper bounds related to various parameters of the graph such as minimum degree and number of edges. We also give upper bounds for random graphs. Last, we give approximability and inapproximability results for general and restricted classes of graphs, and establish a FPT algorithm for interval graphs.Comment: 19 pages, 4 figure

Parameterized Domination in Circle Graphs

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution. Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs. If T is a given tree, deciding whether a circle graph has a dominating set isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by |V(T)|. We prove that the FPT algorithm is subexponential

The Firebreak Problem

Suppose we have a network that is represented by a graph $G$. Potentially a fire (or other type of contagion) might erupt at some vertex of $G$. We are able to respond to this outbreak by establishing a firebreak at $k$ other vertices of $G$, so that the fire cannot pass through these fortified vertices. The question that now arises is which $k$ vertices will result in the greatest number of vertices being saved from the fire, assuming that the fire will spread to every vertex that is not fully behind the $k$ vertices of the firebreak. This is the essence of the {\sc Firebreak} decision problem, which is the focus of this paper. We establish that the problem is intractable on the class of split graphs as well as on the class of bipartite graphs, but can be solved in linear time when restricted to graphs having constant-bounded treewidth, or in polynomial time when restricted to intersection graphs. We also consider some closely related problems