130,202 research outputs found
Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs
A path in an edge-colored graph is rainbow if no two edges of it are
colored the same. The graph is rainbow-connected if there is a rainbow path
between every pair of vertices. If there is a rainbow shortest path between
every pair of vertices, the graph is strongly rainbow-connected. The
minimum number of colors needed to make rainbow-connected is known as the
rainbow connection number of , and is denoted by . Similarly,
the minimum number of colors needed to make strongly rainbow-connected is
known as the strong rainbow connection number of , and is denoted by
. We prove that for every , deciding whether
is NP-complete for split graphs, which form a subclass
of chordal graphs. Furthermore, there exists no polynomial-time algorithm for
approximating the strong rainbow connection number of an -vertex split graph
with a factor of for any unless P = NP. We
then turn our attention to block graphs, which also form a subclass of chordal
graphs. We determine the strong rainbow connection number of block graphs, and
show it can be computed in linear time. Finally, we provide a polynomial-time
characterization of bridgeless block graphs with rainbow connection number at
most 4.Comment: 13 pages, 3 figure
A Characterization of Uniquely Representable Graphs
The betweenness structure of a finite metric space is a pair
where is the so-called betweenness
relation of that consists of point triplets such that . The underlying graph of a betweenness structure
is the simple graph where
the edges are pairs of distinct points with no third point between them. A
connected graph is uniquely representable if there exists a unique metric
betweenness structure with underlying graph . It was implied by previous
works that trees are uniquely representable. In this paper, we give a
characterization of uniquely representable graphs by showing that they are
exactly the block graphs. Further, we prove that two related classes of graphs
coincide with the class of block graphs and the class of distance-hereditary
graphs, respectively. We show that our results hold not only for metric but
also for almost-metric betweenness structures.Comment: 16 pages (without references); 3 figures; major changes: simplified
proofs, improved notations and namings, short overview of metric graph theor
Exploring structural properties of -trees and block graphs
We present a new characterization of -trees based on their reduced clique
graphs and -line graphs, which are block graphs. We explore structural
properties of these two classes, showing that the number of clique-trees of a
-tree equals the number of spanning trees of the -line graph of
. This relationship allows to present a new approach for determining the
number of spanning trees of any connected block graph. We show that these
results can be accomplished in linear time complexity.Comment: 6 pages, 1 figur
Unique Perfect Phylogeny Characterizations via Uniquely Representable Chordal Graphs
The perfect phylogeny problem is a classic problem in computational biology,
where we seek an unrooted phylogeny that is compatible with a set of
qualitative characters. Such a tree exists precisely when an intersection graph
associated with the character set, called the partition intersection graph, can
be triangulated using a restricted set of fill edges. Semple and Steel used the
partition intersection graph to characterize when a character set has a unique
perfect phylogeny. Bordewich, Huber, and Semple showed how to use the partition
intersection graph to find a maximum compatible set of characters. In this
paper, we build on these results, characterizing when a unique perfect
phylogeny exists for a subset of partial characters. Our characterization is
stated in terms of minimal triangulations of the partition intersection graph
that are uniquely representable, also known as ur-chordal graphs. Our
characterization is motivated by the structure of ur-chordal graphs, and the
fact that the block structure of minimal triangulations is mirrored in the
graph that has been triangulated
- …