Log-space Algorithms for Paths and Matchings in k-trees

Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 [JT07]. However, for graphs of tree-width larger than 2, no bound better than NL is known. In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect macthing (decision version), and if so, finding a perfect match- ing (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs [DKR08]. Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of divide-and-conquer approach in log-space, along with some ideas from [JT07] and [LMR07].Comment: Accepted in STACS 201

{\Gamma}-species and the enumeration of k-trees

We study the class of graphs known as k-trees through the lens of Joyal's theory of combinatorial species (and an equivariant extension known as '$\Gamma$-species' which incorporates data about 'structural' group actions). This culminates in a system of recursive functional equations giving the generating function for unlabeled k-trees which allows for fast, efficient computation of their numbers. Enumerations up to k = 10 and n = 30 (for a k-tree with (n+k-1) vertices) are included in tables, and Sage code for the general computation is included in an appendix.Comment: 26 pages; includes Python cod

Canonizing Graphs of Bounded Tree Width in Logspace

Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized by logarithmic-space (logspace) algorithms. This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous class of graphs of bounded tree width one of the few classes of graphs for which the complexity of the isomorphism problem has been exactly determined.Comment: 26 page

{\Gamma}-species, quotients, and graph enumeration

The theory of {\Gamma}-species is developed to allow species-theoretic study of quotient structures in a categorically rigorous fashion. This new approach is then applied to two graph-enumeration problems which were previously unsolved in the unlabeled case-bipartite blocks and general k-trees.Comment: 84 pages, 10 figures, dissertatio

The isomorphism problem for k-trees is complete for logspace

We show that, for k constant, k -tree isomorphism can be decided in logarithmic space by giving an View the MathML sourceO(klogn) space canonical labeling algorithm. The algorithm computes a unique tree decomposition, uses colors to fully encode the structure of the original graph in the decomposition tree and invokes Lindellʼs tree canonization algorithm. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for k -trees are all complete for deterministic logspace. Completeness for logspace holds even for simple structural properties of k -trees. We also show that a variant of our canonical labeling algorithm runs in time View the MathML sourceO((k+1)!n), where n is the number of vertices, yielding the fastest known FPT algorithm for k-tree isomorphism.by Bireswar Das et al.

The isomorphism problem for k-trees is complete for logspace

We show that k-tree isomorphism can be decided in logarithmic space by giving a logspace canonical labeling algorithm. This improves over the previous StUL upper bound and matches the lower bound. As a consequence, the isomorphism, the automorphism, as well as the canonization problem for k-trees are all complete for deterministic logspace. We also show that even simple structural properties of k-trees are complete for logspace