Chuan-Min Lee

New Algorithms for k-Face Cover, k-Feedback Vertex Set, and k -Disjoint Cycles on Plane and Planar Graphs

We present new fixed parameter algorithms for the FACE COVER problem on plane graphs. We show that if a plane graph has a face cover with at most k faces then its treewidth is bounded by O(?k). An approximate tree decomposition can be obtained in linear time, and this is used to find an algorithm computing the face cover number in time O(c?kn) for some constant c. Next we show that the problem is in linear time reducible to a problem kernel of O(k2) vertices, and this kernel can be used to obtain an algorithm that runs in time O(c?k +n) for some other constant c. For the k-DISJOINT CYCLES PROBLEM AND THE k- FEEDBACK VERTEX SET problem on planar graphs we obtain algorithms that run in time O(c?k log kn) for some constant c. For the k-FEEDBACK VERTEX SET problem we can further reduce the problem to a problem kernel of size O(k3) and obtain an algorithm that runs in time O(c?k log k+n) for some constant c1.

Kernels in planar digraphs

A set S of vertices in a digraph D=(V,A) is a kernel if S is independent and every vertex in V-S has an out-neighbor in S. We show that there exist O(n2^1^9^.^1^k+n^4)-time and O(k^3^6+2^1^9^.^1^kk^9+n^2)-time algorithms for checking whether a planar digraph D of order n has a kernel with at most k vertices. Moreover, if D has a kernel of size at most k, the algorithms find such a kernel of minimal size.

Maximum Clique Transversals

A maximum clique transversal set in a graph G is a set S of vertices such that every maximum clique of G contains at least a vertex in S. Clearly, removing a maximum clique transversal set reduces the clique number of a graph. We study algorithmic aspects of the problem, given a graph, to find a maximum clique transversal set of minimum cardinality. We consider the problem for planar graphs and present fixed parameter and approximation results.We also examine some other graph classes: subclasses of chordal graphs such as k-trees, strongly chordal graphs, etc., graphs with few P4s, comparability graphs, and distance hereditary graphs.

An improved algorithm for the maximum agreement subtree problem

In this paper, we solve the maximum agreement subtree problem for a set T of k rooted, leaf-labelled evolutionary trees on n leaves where T contains a binary tree. We show that the O(kn/sup 3/)-time dynamic programming algorithm proposed by Farach et al. and Bryant can be implemented in O(n/sup 2/log/sup k-1/n) and O(k/spl middot/n/sup 3-(1/k-1)/) using the k-dimensional binary search tree and the k-dimensional range search tree, respectively.

On the Recognition of General Partition Graphs

A graph G is a general partition graph if there is some set S and an assignment of non-empty subsets S x ⊆ S to the vertices of G such that two vertices x and y are adjacent if and only if S x ∩ S y ≠ O and for every maximal independent set M the set {S m | m ∈ M} is a partition of S. For every minor closed family of graphs there exists a polynomial time algorithm that checks if an element of the family is a general partition graph.

On the complexity of signed and minus total domination in graphs

In this paper we present unified methods to solve the minus and signed total domination problems for chordal bipartite graphs and trees in O(n^2) and O(n+m) time, respectively. We also prove that the decision problem for the signed total domination problem on doubly chordal graphs is NP-complete. Note that bipartite permutation graphs, biconvex bipartite graphs, and convex bipartite graphs are subclasses of chordal bipartite graphs.

Improved bottleneck domination algorithms

W.C.K. Yen introduced BOTTLENECK DOMINATION and BOTTLENECK INDEPENDENT DOMINATION. He presented an O(n log n + m)- time algorithm to compute a minimum bottleneck dominating set. He also obtained that the BOTTLENECK INDEPENDENT DOMINATING SET problem is NP-complete, even when restricted to planar graphs.We present simple linear time algorithms for the BOTTLENECK DOMINATING SET and the BOTTLENECK TOTAL DOMINATING SET problem. Furthermore, we give polynomial time algorithms (most of them with linear time-complexities) for the BOTTLENECK INDEPENDENT DOMINATING SET problem on the following graph classes: AT-free graphs, chordal graphs, split graphs, permutation graphs, graphs of bounded treewidth, and graphs of clique-width at most k with a given k-expression.