745 research outputs found
Finiteness conditions for graph algebras over tropical semirings
Connection matrices for graph parameters with values in a field have been
introduced by M. Freedman, L. Lov{\'a}sz and A. Schrijver (2007). Graph
parameters with connection matrices of finite rank can be computed in
polynomial time on graph classes of bounded tree-width. We introduce join
matrices, a generalization of connection matrices, and allow graph parameters
to take values in the tropical rings (max-plus algebras) over the real numbers.
We show that rank-finiteness of join matrices implies that these graph
parameters can be computed in polynomial time on graph classes of bounded
clique-width. In the case of graph parameters with values in arbitrary
commutative semirings, this remains true for graph classes of bounded linear
clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that
definability of a graph parameter in Monadic Second Order Logic implies rank
finiteness. We also show that there are uncountably many integer valued graph
parameters with connection matrices or join matrices of fixed finite rank. This
shows that rank finiteness is a much weaker assumption than any definability
assumption.Comment: 12 pages, accepted for presentation at FPSAC 2014 (Chicago, June 29
-July 3, 2014), to appear in Discrete Mathematics and Theoretical Computer
Scienc
Linear rank-width and linear clique-width of trees
We show that for every forest T the linear rank-width of T is equal to the path-width of T, and the linear clique-width of T equals the path-width of T plus two, provided that T contains a path of length three. It follows that both linear rank-width and linear clique-width of forests can be computed in linear time. Using our characterization of linear rank-width of forests, we determine the set of minimal excluded acyclic vertex-minors for the class of graphs of linear rank-width at most k
Faster Algorithms For Vertex Partitioning Problems Parameterized by Clique-width
Many NP-hard problems, such as Dominating Set, are FPT parameterized by
clique-width. For graphs of clique-width given with a -expression,
Dominating Set can be solved in time. However, no FPT algorithm
is known for computing an optimal -expression. For a graph of clique-width
, if we rely on known algorithms to compute a -expression via
rank-width and then solving Dominating Set using the -expression,
the above algorithm will only give a runtime of . There
have been results which overcome this exponential jump; the best known
algorithm can solve Dominating Set in time by avoiding
constructing a -expression [Bui-Xuan, Telle, and Vatshelle. Fast dynamic
programming for locally checkable vertex subset and vertex partitioning
problems. Theoret. Comput. Sci., 2013. doi:10.1016/j.tcs.2013.01.009]. We
improve this to . Indeed, we show that for a graph of
clique-width , a large class of domination and partitioning problems
(LC-VSP), including Dominating Set, can be solved in . Our main tool is a variant of rank-width using the rank of a -
matrix over the rational field instead of the binary field.Comment: 13 pages, 5 figure
An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion
Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and
Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B,
96(4):514--528, 2006]. Motivated from recent development on graph modification
problems regarding classes of graphs of bounded treewidth or pathwidth, we
study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex
Deletion). In the LRW1-Vertex Deletion problem, given an -vertex graph
and a positive integer , we want to decide whether there is a set of at most
vertices whose removal turns into a graph of linear rankwidth at most
and find such a vertex set if one exists. While the meta-theorem of
Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved
in time for some function , it is not clear whether this
problem allows a running time with a modest exponential function.
We first establish that LRW1-Vertex Deletion can be solved in time . The major obstacle to this end is how to handle a long
induced cycle as an obstruction. To fix this issue, we define necklace graphs
and investigate their structural properties. Later, we reduce the polynomial
factor by refining the trivial branching step based on a cliquewidth expression
of a graph, and obtain an algorithm that runs in time . We also prove that the running time cannot be improved to under the Exponential Time Hypothesis assumption. Lastly,
we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.Comment: 29 pages, 9 figures, An extended abstract appeared in IPEC201
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