Domination and Independent Domination on Probe Interval Graphs Ton Kloks, Chih-Shan Liu, Sheng-Lung Peng * Department of Computer Science and Information Engineering, National Dong Hwa University, Hualien 974, Taiwan Abstract A probe interval graph G is a graph of which the vertices are partitioned into a set P of probes and an independent set N of nonprobes such that G can be embedded into an interval graph by adding some edges between vertices of N. In this paper we propose polynomial-time algorithms using dy- namic programming for solving the domination and the independent domination problems on probe interval graphs. 1 Introduction Probe interval graphs, PIGs, were introduced in [17, 18, 19] to model certain problems in phys- ical mapping of DNA. The application in molec- ular biology is the problem of reconstructing the arrangement of fragments of DNA taken from mul- tiple copies of the same genome. The results of laboratory tests tell us which pairs of fragments occupy intersecting intervals of the genome. The genetic information is physically organized in a linear arrangement, and, when full information is available, an arrangement of intervals can be cre- ated in linear time [1]. More recently, a variant that makes more effi- cient use of laboratory resources has been studied. A subset of the fragments is designated as probes, and for each probe one may test all nonprobe frag- ments for intersection with the probe. In graph–theoretic terms, the input to the prob- lem is a graph G and a subset of probe vertices. The other vertices, the nonprobes, form an inde- pendent set in G. The objective is to add edges between certain nonprobe vertices such that the graph becomes an interval graph. This prob- lem has been solved successfully by Johnson and * Corresponding author. E-mail: lung@csie.ndhu.edu.tw Spinrad [15] in O(n 2 ) time (where n is the num- ber of vertices of the graph). Soon after, using modular decomposition, McConnell and Spinrad showed that, alternatively, this could be solved in O(n + m log n) time (where m is the number of edges of the graph) [16]. The problem of recogniz- ing PIGs when the partition of the vertex set is not part of the input was recently solved in [3]. In this paper we give the first polynomial-time algorithms for solving the domination and inde- pendent domination problems for the class of probe interval graphs. 2 Preliminaries Let G =(V, E) be a simple and undirected graph where V and E are the vertex and edge sets of G, respectively. We write n = |V | for the number of vertices and m = |E| for the number of edges. For our convenience we denote an edge as (x, y) rather than {x, y} and say that x and y are adjacent in G. If x and y are adjacent we say that they are the endvertices of the edge. We only deal with graphs that are finite , i.e., the set of vertices is finite. The complement of a graph G, denoted as G, is the graph in which two vertices are adjacent exactly when they are not adjacent in G. For a vertex x we write N(x)= {y ∈ V | (x, y) ∈ E} for the set of neighbors of x in G, and for a subset W ⊆ V we write N(W)= ∪ x∈W N(x). For a vertex x we write N[x]= N(x)+ x for its closed neighborhood. For two sets A and B of elements of a common universe, we write A + B and A - B instead of A ∪ B and A \ B respectively. For a set A and an element x we write A - x instead of A \{x} and A + x instead of A ∪ {x}. For a graph G =(V, E) and a subset S ⊆ V of ver- tices, we write G[S] for the subgraph of G induced by S. For a subset W ⊆ V of vertices of a graph ~93~ The 23rd Workshop on Combinatorial Mathematics and Computation Theory