4,891 research outputs found
Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces
The Subgraph Isomorphism problem asks, given a host graph G on n vertices and
a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P.
The restriction of this problem to planar graphs has often been considered.
After a sequence of improvements, the current best algorithm for planar graphs
is a linear time algorithm by Dorn (STACS '10), with complexity .
We generalize this result, by giving an algorithm of the same complexity for
graphs that can be embedded in surfaces of bounded genus. At the same time, we
simplify the algorithm and analysis. The key to these improvements is the
introduction of surface split decompositions for bounded genus graphs, which
generalize sphere cut decompositions for planar graphs. We extend the algorithm
for the problem of counting and generating all subgraphs isomorphic to P, even
for the case where P is disconnected. This answers an open question by Eppstein
(SODA '95 / JGAA '99)
Dynamic Programming for Graphs on Surfaces
We provide a framework for the design and analysis of dynamic programming
algorithms for surface-embedded graphs on n vertices and branchwidth at most k.
Our technique applies to general families of problems where standard dynamic
programming runs in 2^{O(k log k)} n steps. Our approach combines tools from
topological graph theory and analytic combinatorics. In particular, we
introduce a new type of branch decomposition called "surface cut
decomposition", generalizing sphere cut decompositions of planar graphs
introduced by Seymour and Thomas, which has nice combinatorial properties.
Namely, the number of partial solutions that can be arranged on a surface cut
decomposition can be upper-bounded by the number of non-crossing partitions on
surfaces with boundary. It follows that partial solutions can be represented by
a single-exponential (in the branchwidth k) number of configurations. This
proves that, when applied on surface cut decompositions, dynamic programming
runs in 2^{O(k)} n steps. That way, we considerably extend the class of
problems that can be solved in running times with a single-exponential
dependence on branchwidth and unify/improve most previous results in this
direction.Comment: 28 pages, 3 figure
On Directed Feedback Vertex Set parameterized by treewidth
We study the Directed Feedback Vertex Set problem parameterized by the
treewidth of the input graph. We prove that unless the Exponential Time
Hypothesis fails, the problem cannot be solved in time on general directed graphs, where is the treewidth of
the underlying undirected graph. This is matched by a dynamic programming
algorithm with running time .
On the other hand, we show that if the input digraph is planar, then the
running time can be improved to .Comment: 20
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