4,447 research outputs found
Bounds on monotone switching networks for directed connectivity
We separate monotone analogues of L and NL by proving that any monotone
switching network solving directed connectivity on vertices must have size
at least .Comment: 49 pages, 12 figure
Universal Cycles of Restricted Words
A connected digraph in which the in-degree of any vertex equals its
out-degree is Eulerian, this baseline result is used as the basis of existence
proofs for universal cycles (also known as generalized deBruijn cycles or
U-cycles) of several combinatorial objects. We extend the body of known results
by presenting new results on the existence of universal cycles of monotone,
"augmented onto", and Lipschitz functions in addition to universal cycles of
certain types of lattice paths and random walks.Comment: 21 pages, 4 figure
Counting Euler Tours in Undirected Bounded Treewidth Graphs
We show that counting Euler tours in undirected bounded tree-width graphs is
tractable even in parallel - by proving a upper bound. This is in
stark contrast to #P-completeness of the same problem in general graphs.
Our main technical contribution is to show how (an instance of) dynamic
programming on bounded \emph{clique-width} graphs can be performed efficiently
in parallel. Thus we show that the sequential result of Espelage, Gurski and
Wanke for efficiently computing Hamiltonian paths in bounded clique-width
graphs can be adapted in the parallel setting to count the number of
Hamiltonian paths which in turn is a tool for counting the number of Euler
tours in bounded tree-width graphs. Our technique also yields parallel
algorithms for counting longest paths and bipartite perfect matchings in
bounded-clique width graphs.
While establishing that counting Euler tours in bounded tree-width graphs can
be computed by non-uniform monotone arithmetic circuits of polynomial degree
(which characterize ) is relatively easy, establishing a uniform
bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound
claimed in the previous version which has been subsequently remove
Induced minors and well-quasi-ordering
A graph is an induced minor of a graph if it can be obtained from an
induced subgraph of by contracting edges. Otherwise, is said to be
-induced minor-free. Robin Thomas showed that -induced minor-free
graphs are well-quasi-ordered by induced minors [Graphs without and
well-quasi-ordering, Journal of Combinatorial Theory, Series B, 38(3):240 --
247, 1985].
We provide a dichotomy theorem for -induced minor-free graphs and show
that the class of -induced minor-free graphs is well-quasi-ordered by the
induced minor relation if and only if is an induced minor of the gem (the
path on 4 vertices plus a dominating vertex) or of the graph obtained by adding
a vertex of degree 2 to the complete graph on 4 vertices. To this end we proved
two decomposition theorems which are of independent interest.
Similar dichotomy results were previously given for subgraphs by Guoli Ding
in [Subgraphs and well-quasi-ordering, Journal of Graph Theory, 16(5):489--502,
1992] and for induced subgraphs by Peter Damaschke in [Induced subgraphs and
well-quasi-ordering, Journal of Graph Theory, 14(4):427--435, 1990]
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