1,509 research outputs found
Decomposing Cubic Graphs into Connected Subgraphs of Size Three
Let be the set of connected graphs of size 3. We
study the problem of partitioning the edge set of a graph into graphs taken
from any non-empty . The problem is known to be NP-complete for
any possible choice of in general graphs. In this paper, we assume that
the input graph is cubic, and study the computational complexity of the problem
of partitioning its edge set for any choice of . We identify all polynomial
and NP-complete problems in that setting, and give graph-theoretic
characterisations of -decomposable cubic graphs in some cases.Comment: to appear in the proceedings of COCOON 201
Towards an Isomorphism Dichotomy for Hereditary Graph Classes
In this paper we resolve the complexity of the isomorphism problem on all but
finitely many of the graph classes characterized by two forbidden induced
subgraphs. To this end we develop new techniques applicable for the structural
and algorithmic analysis of graphs. First, we develop a methodology to show
isomorphism completeness of the isomorphism problem on graph classes by
providing a general framework unifying various reduction techniques. Second, we
generalize the concept of the modular decomposition to colored graphs, allowing
for non-standard decompositions. We show that, given a suitable decomposition
functor, the graph isomorphism problem reduces to checking isomorphism of
colored prime graphs. Third, we extend the techniques of bounded color valence
and hypergraph isomorphism on hypergraphs of bounded color size as follows. We
say a colored graph has generalized color valence at most k if, after removing
all vertices in color classes of size at most k, for each color class C every
vertex has at most k neighbors in C or at most k non-neighbors in C. We show
that isomorphism of graphs of bounded generalized color valence can be solved
in polynomial time.Comment: 37 pages, 4 figure
Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs
The Graph Isomorphism problem restricted to graphs of bounded treewidth or
bounded tree distance width are known to be solvable in polynomial time
[Bod90],[YBFT99]. We give restricted space algorithms for these problems
proving the following results: - Isomorphism for bounded tree distance width
graphs is in L and thus complete for the class. We also show that for this kind
of graphs a canon can be computed within logspace. - For bounded treewidth
graphs, when both input graphs are given together with a tree decomposition,
the problem of whether there is an isomorphism which respects the
decompositions (i.e. considering only isomorphisms mapping bags in one
decomposition blockwise onto bags in the other decomposition) is in L. - For
bounded treewidth graphs, when one of the input graphs is given with a tree
decomposition the isomorphism problem is in LogCFL. - As a corollary the
isomorphism problem for bounded treewidth graphs is in LogCFL. This improves
the known TC1 upper bound for the problem given by Grohe and Verbitsky
[GroVer06].Comment: STACS conference 2010, 12 page
Transversal designs and induced decompositions of graphs
We prove that for every complete multipartite graph there exist very
dense graphs on vertices, namely with as many as
edges for all , for some constant , such that can be
decomposed into edge-disjoint induced subgraphs isomorphic to~. This result
identifies and structurally explains a gap between the growth rates and
on the minimum number of non-edges in graphs admitting an
induced -decomposition
Rank-width and Tree-width of H-minor-free Graphs
We prove that for any fixed r>=2, the tree-width of graphs not containing K_r
as a topological minor (resp. as a subgraph) is bounded by a linear (resp.
polynomial) function of their rank-width. We also present refinements of our
bounds for other graph classes such as K_r-minor free graphs and graphs of
bounded genus.Comment: 17 page
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