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

Parameterized complexity of coloring problems: Treewidth versus vertex cover

AbstractWe compare the fixed parameter complexity of various variants of coloring problems (including List Coloring, Precoloring Extension, Equitable Coloring, L(p,1)-Labeling and Channel Assignment) when parameterized by treewidth and by vertex cover number. In most (but not all) cases we conclude that parametrization by the vertex cover number provides a significant drop in the complexity of the problems

Co-Degeneracy and Co-Treewidth: Using the Complement to Solve Dense Instances

Clique-width and treewidth are two of the most important and useful graph parameters, and several problems can be solved efficiently when restricted to graphs of bounded clique-width or treewidth. Bounded treewidth implies bounded clique-width, but not vice versa. Problems like Longest Cycle, Longest Path, MaxCut, Edge Dominating Set, and Graph Coloring are fixed-parameter tractable when parameterized by the treewidth, but they cannot be solved in FPT time when parameterized by the clique-width unless FPT = W[1], as shown by Fomin, Golovach, Lokshtanov, and Saurabh [SIAM J. Comput. 2010, SIAM J. Comput. 2014]. For a given problem that is fixed-parameter tractable when parameterized by treewidth, but intractable when parameterized by clique-width, there may exist infinite families of instances of bounded clique-width and unbounded treewidth where the problem can be solved efficiently. In this work, we initiate a systematic study of the parameters co-treewidth (the treewidth of the complement of the input graph) and co-degeneracy (the degeneracy of the complement of the input graph). We show that Longest Cycle, Longest Path, and Edge Dominating Set are FPT when parameterized by co-degeneracy. On the other hand, Graph Coloring is para-NP-complete when parameterized by co-degeneracy but FPT when parameterized by the co-treewidth. Concerning MaxCut, we give an FPT algorithm parameterized by co-treewidth, while we leave open the complexity of the problem parameterized by co-degeneracy. Additionally, we show that Precoloring Extension is fixed-parameter tractable when parameterized by co-treewidth, while this problem is known to be W[1]-hard when parameterized by treewidth. These results give evidence that co-treewidth is a useful width parameter for handling dense instances of problems for which an FPT algorithm for clique-width is unlikely to exist. Finally, we develop an algorithmic framework for co-degeneracy based on the notion of Bondy-Chvátal closure.publishedVersio

Mixed graph edge coloring

We are interested in coloring the edges of a mixed graph, i.e., a graph containing unoriented and oriented edges. This problem is related to a communication problem in job-shop scheduling systems. In this paper we give general bounds on the number of required colors and analyze the complexity status of this problem. In particular, we provide NP-completeness results for the case of outerplanar graphs, as well as for 3- regular bipartite graphs (even when only 3 colors are allowed, or when 5 colors are allowed and the graph is fully oriented). Special cases admitting polynomial-time solutions are also discussed

On the complexity of some colorful problems parameterized by treewidth

In this paper,we study the complexity of several coloring problems on graphs, parameterizedby the treewidth of the graph.1. The List Coloring problem takes as input a graph G, togetherwith an assignment to each vertex v of a set of colors Cv. The problem is to determinewhether it is possible to choose a color for vertex v from the set of permitted colors Cv, for each vertex, so that the obtained coloring of G is proper. We show that this problem is W[1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W[1]-hard, parameterized by treewidth.2. An equitable coloring of a graph G is a proper coloring of the verticeswhere the numbers of vertices having any two distinct colors differs by at most one.We show that the problem is hard forW[1], parameterized by the treewidth plus the number of colors.We also show that a list-based variation, List Equitable Coloring is W[1]-hard for forests, parameterizedby the number of colors on the lists.3. The list chromatic number χl(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list Lv of colors, where each list has length at least r, there is a choice of one color fromeach vertex list Lv yielding a proper coloring of G. We show that the problem of determining whether χl(G) ≤ r, the ListChromatic Number problem, is solvable in linear time on graphs of constant treewidth

List-coloring and sum-list-coloring problems on graphs

Graph coloring is a well-known and well-studied area of graph theory that has many applications. In this dissertation, we look at two generalizations of graph coloring known as list-coloring and sum-list-coloring. In both of these types of colorings, one seeks to first assign palettes of colors to vertices and then choose a color from the corresponding palette for each vertex so that a proper coloring is obtained. A celebrated result of Thomassen states that every planar graph can be properly colored from any arbitrarily assigned palettes of five colors. This result is known as 5-list-colorability of planar graphs. Albertson asked whether Thomassen\u27s theorem can be extended by precoloring some vertices which are at a large enough distance apart. Hutchinson asked whether Thomassen\u27s theorem can be extended by allowing certain vertices to have palettes of size less than five assigned to them. In this dissertation, we explore both of these questions and answer them in the affirmative for various classes of graphs. We also provide a catalog of small configurations with palettes of different prescribed sizes and determine whether or not they can always be colored from palettes of such sizes. These small configurations can be useful in reducing certain planar graphs to obtain more information about their structure. Additionally, we look at the newer notion of sum-list-coloring where the sum choice number is the parameter of interest. In sum-list-coloring, we seek to minimize the sum of varying sizes of palettes of colors assigned the vertices of a graph. We compute the sum choice number for all graphs on at most five vertices, present some general results about sum-list-coloring, and determine the sum choice number for certain graphs made up of cycles