1,533 research outputs found
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
Characterising and recognising game-perfect graphs
Consider a vertex colouring game played on a simple graph with
permissible colours. Two players, a maker and a breaker, take turns to colour
an uncoloured vertex such that adjacent vertices receive different colours. The
game ends once the graph is fully coloured, in which case the maker wins, or
the graph can no longer be fully coloured, in which case the breaker wins. In
the game , the breaker makes the first move. Our main focus is on the
class of -perfect graphs: graphs such that for every induced subgraph ,
the game played on admits a winning strategy for the maker with only
colours, where denotes the clique number of .
Complementing analogous results for other variations of the game, we
characterise -perfect graphs in two ways, by forbidden induced subgraphs
and by explicit structural descriptions. We also present a clique module
decomposition, which may be of independent interest, that allows us to
efficiently recognise -perfect graphs.Comment: 39 pages, 8 figures. An extended abstract was accepted at the
International Colloquium on Graph Theory (ICGT) 201
On graphs with no induced subdivision of
We prove a decomposition theorem for graphs that do not contain a subdivision
of as an induced subgraph where is the complete graph on four
vertices. We obtain also a structure theorem for the class of graphs
that contain neither a subdivision of nor a wheel as an induced subgraph,
where a wheel is a cycle on at least four vertices together with a vertex that
has at least three neighbors on the cycle. Our structure theorem is used to
prove that every graph in is 3-colorable and entails a polynomial-time
recognition algorithm for membership in . As an intermediate result, we
prove a structure theorem for the graphs whose cycles are all chordless
Embedability between right-angled Artin groups
In this article we study the right-angled Artin subgroups of a given
right-angled Artin group. Starting with a graph \gam, we produce a new graph
through a purely combinatorial procedure, and call it the extension graph
\gam^e of \gam. We produce a second graph \gam^e_k, the clique graph of
\gam^e, by adding extra vertices for each complete subgraph of \gam^e. We
prove that each finite induced subgraph of \gam^e gives rise to an
inclusion A(\Lambda)\to A(\gam). Conversely, we show that if there is an
inclusion A(\Lambda)\to A(\gam) then is an induced subgraph of
\gam^e_k. These results have a number of corollaries. Let denote the
path on four vertices and let denote the cycle of length . We prove
that embeds in A(\gam) if and only if is an induced subgraph
of \gam. We prove that if is any finite forest then embeds in
. We recover the first author's result on co--contraction of graphs and
prove that if \gam has no triangles and A(\gam) contains a copy of
for some , then \gam contains a copy of for some . We also recover Kambites' Theorem, which asserts that if embeds in
A(\gam) then \gam contains an induced square. Finally, we determine
precisely when there is an inclusion and show that there is
no "universal" two--dimensional right-angled Artin group.Comment: 35 pages. Added an appendix and a proof that the extension graph is
quasi-isometric to a tre
Completing Partial Packings of Bipartite Graphs
Given a bipartite graph and an integer , let be the smallest
integer such that, any set of edge disjoint copies of on vertices, can
be extended to an -design on at most vertices. We establish tight
bounds for the growth of as . In particular, we
prove the conjecture of F\"uredi and Lehel \cite{FuLe} that .
This settles a long-standing open problem
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