A refined search tree technique for Dominating Set on planar graphs

Abstract

We establish a refined search tree technique for the parameterized DOMINATING SET problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that every other vertex has at least one neighbor in this set. We describe algorithms with running times O(8 k n) and O(8 k k + n 3 ), where n is the number of vertices in the graph, based on bounded search trees. We describe a set of polynomial time data-reduction rules for a more general "annotated" problem on black/white graphs that asks for a set of k vertices (black or white) ଁ An extended abstract of this paper appeared under nearly the same title in that dominate all the black vertices. An intricate argument based on the Euler formula then establishes an efficient branching strategy for reduced inputs to this problem. In addition, we give a family examples showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final search tree algorithm is easy to implement; its analysis, however, is involved.

An Experimental Study on Generating Planar Graphs

“…When looking at the slope of the histogram, we can see that more than 99.6% of the faces of the generated graphs have a size within the interval [3,10]. The slope has its peak at size 3 and decreases fast with growing face size.…”

“…Kuratowski. The Kuratowski generator is based on the Kuratowski theorem, which states that a graph is planar if and only if no subgraph is present that is a subdivision of K 5 or K 3,3 . The generator starts with a non-planar random graph.…”

“…However, the problem reduction of PDS does not rely on the degree of the nodes but the neighborhood of an examined node. The algorithms described in [4] and [3] have been implemented. The reduction is based on two kernelization and seven search tree rules.…”

Short Cycles Make W-hard Problems Hard: FPT Algorithms for W-hard Problems in Graphs with no Short Cycles

“…Though variations of Dominating Set like Red-Blue Dominating Set [10] and Constrained Dominating Set [16] have been studied before and shown to be W [2]-complete, to the best of our knowledge the standard Dominating Set problem (which we consider here) in bipartite graphs has not been studied before. Our observation means that the dominating set problem is W [2]-complete in triangle free graphs.…”

“…We say that the set D "dominates" the vertices of G. We first show that Dominating Set problem is W [2]-complete in bipartite graphs and split graphs by a reduction from the same problem in general undirected graphs. Then we give a fixed parameter tractable algorithm for the problem in graphs with girth at least 5.…”