We present an algorithm that constructively produces a solution to the k-DOMINATING SET problem for planar graphs in time O(c √ k n), where c = 4 6 √ 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O( √ γ (G)), and that such a tree decomposition can be found in O( √ γ (G)n) time. The same technique can be used to show that the k-FACE COVER problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c √ k 1 n) time, where c 1 = 3 36 √ 34and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k-DOMINATING SET, e.g., k-INDEPENDENT DOMINATING SET and k-WEIGHTED DOMINATING SET.

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A v ertex edge coloring c : V ! f 1; 2; : : : ; t g c 0 : E ! f 1; 2; : : : ; t g of a graph G = V ;E i s a v ertex edge t-ranking if for any t wo v ertices edges of the same color every path between them contains a vertex edge of larger color. The vertex ranking number r G edge ranking number 0 r G is the smallest value of t such that G has a vertex edge t-ranking. In this paper we study the algorithmic complexity o f t h e vertex ranking and edge ranking problems. Among others it is shown that r G can be computed in polynomial time when restricted to graphs with treewidth at most k for any xed k. We c haracterize those graphs where the vertex ranking number r and the chromatic number coincide on all induced subgraphs, show that r G = G implies G = !G largest clique size and give a formula for 0 r K n .

We present O(n'R + n3R3) time algorithms to compute the treewidth, pathwidth, minimum fill-in and minimum interval graph completion of asteroidal triple-free graphs, where n is the number of vertices and R is the number of minimal separators of the input graph. This yields polynomial time algorithms for the four NP-complete graph problems on any subclass of the asteroidal triple-free graphs that has a polynomially bounded number of minimal separators, as e.g. cocomparability graphs of bounded dimension and d-trapezoid graphs for any fixed d > 1.