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Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree
The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for
every tree there exists a natural number such that the following
holds: If is a -edge-connected simple graph with size divisible by
the size of , then can be edge-decomposed into subgraphs isomorphic to
. So far this conjecture has only been verified for paths, stars, and a
family of bistars. We prove a weaker version of the Tree Decomposition
Conjecture, where we require the subgraphs in the decomposition to be
isomorphic to graphs that can be obtained from by vertex-identifications.
We call such a subgraph a homomorphic copy of . This implies the Tree
Decomposition Conjecture under the additional constraint that the girth of
is greater than the diameter of . As an application, we verify the Tree
Decomposition Conjecture for all trees of diameter at most 4.Comment: 18 page
Composing dynamic programming tree-decomposition-based algorithms
Given two integers and as well as graph classes
, the problems
,
, and
ask, given graph
as input, whether , , respectively can be partitioned
into sets such that, for each between and
, , , respectively. Moreover in , we request that the number of edges with
endpoints in different sets of the partition is bounded by . We show that if
there exist dynamic programming tree-decomposition-based algorithms for
recognizing the graph classes , for each , then we can
constructively create a dynamic programming tree-decomposition-based algorithms
for ,
, and
. We show that, in
some known cases, the obtained running times are comparable to those of the
best know algorithms
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