255 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
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
Obstructions for bounded shrub-depth and rank-depth
Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph.
It is well known that a graph has large tree-depth if and only if it has a long
path as a subgraph. We prove an analogous statement for shrub-depth and
rank-depth, which was conjectured by Hlin\v{e}n\'y, Kwon, Obdr\v{z}\'alek, and
Ordyniak [Tree-depth and vertex-minors, European J.~Combin. 2016]. Namely, we
prove that a graph has large rank-depth if and only if it has a vertex-minor
isomorphic to a long path. This implies that for every integer , the class
of graphs with no vertex-minor isomorphic to the path on vertices has
bounded shrub-depth.Comment: 19 pages, 5 figures; accepted to Journal of Combinatorial Theory Ser.
Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm
Linear rank-width is a linearized variation of rank-width, and it is deeply
related to matroid path-width. In this paper, we show that the linear
rank-width of every -vertex distance-hereditary graph, equivalently a graph
of rank-width at most , can be computed in time , and a linear layout witnessing the linear rank-width can be computed with
the same time complexity. As a corollary, we show that the path-width of every
-element matroid of branch-width at most can be computed in time
, provided that the matroid is given by an
independent set oracle.
To establish this result, we present a characterization of the linear
rank-width of distance-hereditary graphs in terms of their canonical split
decompositions. This characterization is similar to the known characterization
of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex
separation and search number of a graph. Inf. Comput., 113(1):50--79, 1994].
However, different from forests, it is non-trivial to relate substructures of
the canonical split decomposition of a graph with some substructures of the
given graph. We introduce a notion of `limbs' of canonical split
decompositions, which correspond to certain vertex-minors of the original
graph, for the right characterization.Comment: 28 pages, 3 figures, 2 table. A preliminary version appeared in the
proceedings of WG'1
A tight relation between series--parallel graphs and bipartite distance hereditary graphs
Bandelt and Mulderâs structural characterization of bipartite distance hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by re17 peatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffinâs structural characterization of 2âconnected seriesâparallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we give an elementary proof that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and seriesâparallel graphs and to provide a new class of polynomially-solvable instances for the integer multi-commodity flow of maximum valu
Graph-Links
The present paper is a review of the current state of Graph-Link Theory
(graph-links are also closely related to homotopy classes of looped
interlacement graphs), dealing with a generalisation of knots obtained by
translating the Reidemeister moves for links into the language of intersection
graphs of chord diagrams. In this paper we show how some methods of classical
and virtual knot theory can be translated into the language of abstract graphs,
and some theorems can be reproved and generalised to this graphical setting. We
construct various invariants, prove certain minimality theorems and construct
functorial mappings for graph-knots and graph-links. In this paper, we first
show non-equivalence of some graph-links to virtual links.Comment: 32 pages, 21 figure
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