1,240 research outputs found
Bad News for Chordal Partitions
Reed and Seymour [1998] asked whether every graph has a partition into
induced connected non-empty bipartite subgraphs such that the quotient graph is
chordal. If true, this would have significant ramifications for Hadwiger's
Conjecture. We prove that the answer is `no'. In fact, we show that the answer
is still `no' for several relaxations of the question
On Tree-Partition-Width
A \emph{tree-partition} of a graph is a proper partition of its vertex
set into `bags', such that identifying the vertices in each bag produces a
forest. The \emph{tree-partition-width} of is the minimum number of
vertices in a bag in a tree-partition of . An anonymous referee of the paper
by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph
with tree-width and maximum degree has
tree-partition-width at most . We prove that this bound is within a
constant factor of optimal. In particular, for all and for all
sufficiently large , we construct a graph with tree-width , maximum
degree , and tree-partition-width at least (\eighth-\epsilon)k\Delta.
Moreover, we slightly improve the upper bound to
without the restriction that
Colouring exact distance graphs of chordal graphs
For a graph and positive integer , the exact distance- graph
is the graph with vertex set and with an edge between
vertices and if and only if and have distance . Recently,
there has been an effort to obtain bounds on the chromatic number
of exact distance- graphs for from certain
classes of graphs. In particular, if a graph has tree-width , it has
been shown that for odd ,
and for even . We
show that if is chordal and has tree-width , then for odd , and for even .
If we could show that for every graph of tree-width there is a
chordal graph of tree-width which contains as an isometric subgraph
(i.e., a distance preserving subgraph), then our results would extend to all
graphs of tree-width . While we cannot do this, we show that for every graph
of genus there is a graph which is a triangulation of genus and
contains as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which
arise from reviewers' comment
Dichotomy for tree-structured trigraph list homomorphism problems
Trigraph list homomorphism problems (also known as list matrix partition
problems) have generated recent interest, partly because there are concrete
problems that are not known to be polynomial time solvable or NP-complete. Thus
while digraph list homomorphism problems enjoy dichotomy (each problem is
NP-complete or polynomial time solvable), such dichotomy is not necessarily
expected for trigraph list homomorphism problems. However, in this paper, we
identify a large class of trigraphs for which list homomorphism problems do
exhibit a dichotomy. They consist of trigraphs with a tree-like structure, and,
in particular, include all trigraphs whose underlying graphs are trees. In
fact, we show that for these tree-like trigraphs, the trigraph list
homomorphism problem is polynomially equivalent to a related digraph list
homomorphism problem. We also describe a few examples illustrating that our
conditions defining tree-like trigraphs are not unnatural, as relaxing them may
lead to harder problems
Unique perfect phylogeny is NP-hard
We answer, in the affirmative, the following question proposed by Mike Steel
as a $100 challenge: "Is the following problem NP-hard? Given a ternary
phylogenetic X-tree T and a collection Q of quartet subtrees on X, is T the
only tree that displays Q ?
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
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