2,246 research outputs found
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
A Note on Graphs of Linear Rank-Width 1
We prove that a connected graph has linear rank-width 1 if and only if it is
a distance-hereditary graph and its split decomposition tree is a path. An
immediate consequence is that one can decide in linear time whether a graph has
linear rank-width at most 1, and give an obstruction if not. Other immediate
consequences are several characterisations of graphs of linear rank-width 1. In
particular a connected graph has linear rank-width 1 if and only if it is
locally equivalent to a caterpillar if and only if it is a vertex-minor of a
path [O-joung Kwon and Sang-il Oum, Graphs of small rank-width are pivot-minors
of graphs of small tree-width, arxiv:1203.3606] if and only if it does not
contain the co-K_2 graph, the Net graph and the 5-cycle graph as vertex-minors
[Isolde Adler, Arthur M. Farley and Andrzej Proskurowski, Obstructions for
linear rank-width at most 1, arxiv:1106.2533].Comment: 9 pages, 2 figures. Not to be publishe
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
Oriented coloring on recursively defined digraphs
Coloring is one of the most famous problems in graph theory. The coloring
problem on undirected graphs has been well studied, whereas there are very few
results for coloring problems on directed graphs. An oriented k-coloring of an
oriented graph G=(V,A) is a partition of the vertex set V into k independent
sets such that all the arcs linking two of these subsets have the same
direction. The oriented chromatic number of an oriented graph G is the smallest
k such that G allows an oriented k-coloring. Deciding whether an acyclic
digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the
chromatic number of an oriented graph is an NP-hard problem. This motivates to
consider the problem on oriented co-graphs. After giving several
characterizations for this graph class, we show a linear time algorithm which
computes an optimal oriented coloring for an oriented co-graph. We further
prove how the oriented chromatic number can be computed for the disjoint union
and order composition from the oriented chromatic number of the involved
oriented co-graphs. It turns out that within oriented co-graphs the oriented
chromatic number is equal to the length of a longest oriented path plus one. We
also show that the graph isomorphism problem on oriented co-graphs can be
solved in linear time.Comment: 14 page
Definable decompositions for graphs of bounded linear cliquewidth
We prove that for every positive integer , there exists an
-transduction that given a graph of linear cliquewidth at most
outputs, nondeterministically, some cliquewidth decomposition of the graph
of width bounded by a function of . A direct corollary of this result is the
equivalence of the notions of -definability and recognizability
on graphs of bounded linear cliquewidth.Comment: 39 pages, 5 figures. The conference version of the manuscript
appeared in the proceedings of LICS 201
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