Dominating Sets and Induced Matchings in Orthogonal Ray Graphs

Abstract

Asahi TAKAOKA†a) , Student Member, Satoshi TAYU †b) , Member, and Shuichi UENO †c) , Fellow SUMMARY An orthogonal ray graph is an intersection graph of horizontal and vertical rays (closed half-lines) in the plane. Such a graph is 3-directional if every vertical ray has the same direction, and 2-directional if every vertical ray has the same direction and every horizontal ray has the same direction. We derive some structural properties of orthogonal ray graphs, and based on these properties, we introduce polynomial-time algorithms that solve the dominating set problem, the induced matching problem, and the strong edge coloring problem for these graphs. We show that for 2-directional orthogonal ray graphs, the dominating set problem can be solved in O(n 2 log 5 n) time, the weighted dominating set problem can be solved in O(n 4 log n) time, and the number of dominating sets of a fixed size can be computed in O(n 6 log n) time, where n is the number of vertices in the graph. We also show that for 2-directional orthogonal ray graphs, the weighted induced matching problem and the strong edge coloring problem can be solved in O(n 2 + m log n) time, where m is the number of edges in the graph. Moreover, we show that for 3-directional orthogonal ray graphs, the induced matching problem can be solved in O(m 2 ) time, the weighted induced matching problem can be solved in O(m 4 ) time, and the strong edge coloring problem can be solved in O(m 3 ) time. We finally show that the weighted induced matching problem can be solved in O(m 6 ) time for orthogonal ray graphs.

Dominating Sets in Two-Directional Orthogonal Ray Graphs

“…With this observation, we show in [13] that the weighted dominating set problem can be solved in O(n 4 log n) time for 2-DORGs. We also use it in this paper, and we refer to u S as the representative of S .…”

“…Also, some problems are known to be solvable or approximable in polynomial time for 2-DORGs [5], [7], [8], [11]- [14]. We recently showed in [13] that the weighted dominating set problem can be solved in O(n 4 log n) time for 2-DORGs by using a new parameter, boolean-width of graphs. Boolean-width of graphs is introduced in [2], [3], and several problems can be solved in polynomial time by dynamic programming algorithms if the graphs has boolean-width O(log n).…”

“…We expect that the complexity of the problems can be reduced by using direct dynamic programming approaches, as shown in this paper. It should also be noted that the complexity of the weighted dominating set problem for orthogonal ray graphs still remains open, whereas the problem can be solved in polynomial time provided that orthogonal ray representations of graphs [13], [15].…”

Graph Isomorphism Completeness for Trapezoid Graphs

“…Proof. The complements of comparability graphs of posets with interval dimension at most 2 and height at most 2 are circular-arc graphs with clique-cover number 2 [16,17], for which the graph isomorphism problem can be solved in linear time [5,7]. Since it requires O(n 2 ) time to take the complements of graphs, we have the proposition.…”

“…Comparability graphs of posets with interval dimension at most 2 and height at most 2 are also known as 2-directional orthogonal ray graphs [13,16]. See [19] for more information on the isomorphism of these graphs.…”