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Pathwidth and ThreeDimensional StraightLine Grid Drawings of Graphs
"... We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for ..."
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We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for
LOGSPACE ALGORITHMS FOR PATHS AND MATCHINGS IN kTrees
 STACS
, 2010
"... Reachability and shortest path problems are NLcomplete for general graphs. They are known to be in L for graphs of treewidth 2 [14]. However, for graphs of treewidth larger than 2, no bound better than NL is known. In this paper, we improve these bounds for ktrees, where k is a constant. In parti ..."
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Reachability and shortest path problems are NLcomplete for general graphs. They are known to be in L for graphs of treewidth 2 [14]. However, for graphs of treewidth larger than 2, no bound better than NL is known. In this paper, we improve these bounds for ktrees, where k is a constant. In particular, the main results of our paper are logspace algorithms for reachability in directed ktrees, and for computation of shortest and longest paths in directed acyclic ktrees. Besides the path problems mentioned above, we consider the problem of deciding whether a ktree has a perfect macthing (decision version), and if so, finding a perfect matching (search version), and prove that these problems are Lcomplete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs [8]. Our results settle the complexity of these problems for the class of ktrees. The results are also applicable for bounded treewidth graphs, when a treedecomposition is given as input. The technique central to our algorithms is a careful implementation of divideandconquer approach in logspace, along with some ideas from [14] and [19].