15,423 research outputs found
Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions
In the companion paper [Linear rank-width of distance-hereditary graphs I. A
polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a
characterization of the linear rank-width of distance-hereditary graphs, from
which we derived an algorithm to compute it in polynomial time. In this paper,
we investigate structural properties of distance-hereditary graphs based on
this characterization.
First, we prove that for a fixed tree , every distance-hereditary graph of
sufficiently large linear rank-width contains a vertex-minor isomorphic to .
We extend this property to bigger graph classes, namely, classes of graphs
whose prime induced subgraphs have bounded linear rank-width. Here, prime
graphs are graphs containing no splits. We conjecture that for every tree ,
every graph of sufficiently large linear rank-width contains a vertex-minor
isomorphic to . Our result implies that it is sufficient to prove this
conjecture for prime graphs.
For a class of graphs closed under taking vertex-minors, a graph
is called a vertex-minor obstruction for if but all of
its proper vertex-minors are contained in . Secondly, we provide, for
each , a set of distance-hereditary graphs that contains all
distance-hereditary vertex-minor obstructions for graphs of linear rank-width
at most . Also, we give a simpler way to obtain the known vertex-minor
obstructions for graphs of linear rank-width at most .Comment: 38 pages, 13 figures, 1 table, revised journal version. A preliminary
version of Section 5 appeared in the proceedings of WG1
Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs
A graph is {\em matching-decyclable} if it has a matching such that
is acyclic. Deciding whether is matching-decyclable is an NP-complete
problem even if is 2-connected, planar, and subcubic. In this work we
present results on matching-decyclability in the following classes: Hamiltonian
subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian
subcubic graphs we show that deciding matching-decyclability is NP-complete
even if there are exactly two vertices of degree two. For chordal and
distance-hereditary graphs, we present characterizations of
matching-decyclability that lead to -time recognition algorithms
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
( k , +)-distance-hereditary graphs
AbstractIn this work we introduce, characterize, and provide algorithmic results for (k,+)-distance-hereditary graphs, k⩾0. These graphs can be used to model interconnection networks with desirable connectivity properties; a network modeled as a (k,+)-distance-hereditary graph can be characterized as follows: if some nodes have failed, as long as two nodes remain connected, the distance between these nodes in the faulty graph is bounded by the distance in the non-faulty graph plus an integer constant k. The class of all these graphs is denoted by DH(k,+). By varying the parameter k, classes DH(k,+) include all graphs and form a hierarchy that represents a parametric extension of the well-known class of distance-hereditary graphs
Hamilton cycles in almost distance-hereditary graphs
Let be a graph on vertices. A graph is almost
distance-hereditary if each connected induced subgraph of has the
property for any pair of vertices .
A graph is called 1-heavy (2-heavy) if at least one (two) of the end
vertices of each induced subgraph of isomorphic to (a claw) has
(have) degree at least , and called claw-heavy if each claw of has a
pair of end vertices with degree sum at least . Thus every 2-heavy graph is
claw-heavy. In this paper we prove the following two results: (1) Every
2-connected, claw-heavy and almost distance-hereditary graph is Hamiltonian.
(2) Every 3-connected, 1-heavy and almost distance-hereditary graph is
Hamiltonian. In particular, the first result improves a previous theorem of
Feng and Guo. Both results are sharp in some sense.Comment: 14 pages; 1 figure; a new theorem is adde
A Graph Class related to the Structural Domination Problem
In the structural domination problem one is concerned with the question if a given graph has a connected dominating set whose induced subgraph has certain structural properties. For most of the common graph properties, the associated decision problem is NP-hard. Recently, Bacsô and Tuza gave a full characterization of the graphs whose every induced subgraph has a connected dominating set satisfying an arbitrary prescribed hereditary property. Using the Theorem of Bacsô and Tuza, we derive a finite forbidden subgraph characterization of the distance-hereditary graphs that have a dominating induced tree. Furthermore, we show that for distance-hereditary graphs minimum dominating induced trees can be found efficiently. The main part of the paper studies a new class of graphs, the extit{structural domination class}, which is closely related to the structural domination problem. Among other results, we give new characterizations of certain subclasses of distance-hereditary graphs (in particular for ptolemaic graphs) and analyse the structure of minimum connected dominating sets of structural domination graphs. It turns out that many of the problems associated to structural domination become tractable on the hereditary part of the structural domination class
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