363 research outputs found
Regularity of Edge Ideals and Their Powers
We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals
of graphs and their powers. Our focus is on bounds and exact values of and the asymptotic linear function , for in terms of combinatorial data of the given graph Comment: 31 pages, 15 figure
Even-cycle decompositions of graphs with no odd--minor
An even-cycle decomposition of a graph G is a partition of E(G) into cycles
of even length. Evidently, every Eulerian bipartite graph has an even-cycle
decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian
planar graph with an even number of edges also admits an even-cycle
decomposition. Later, Zhang (1994) generalized this to graphs with no
-minor.
Our main theorem gives sufficient conditions for the existence of even-cycle
decompositions of graphs in the absence of odd minors. Namely, we prove that
every 2-connected loopless Eulerian odd--minor-free graph with an even
number of edges has an even-cycle decomposition.
This is best possible in the sense that `odd--minor-free' cannot be
replaced with `odd--minor-free.' The main technical ingredient is a
structural characterization of the class of odd--minor-free graphs, which
is due to Lov\'asz, Seymour, Schrijver, and Truemper.Comment: 17 pages, 6 figures; minor revisio
Isolation Schemes for Problems on Decomposable Graphs
The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much effort has been dedicated towards derandomization of the Isolation Lemma for specific classes of problems. So far, the focus was mainly on problems solvable in polynomial time. In this paper, we study a setting that is more typical for -complete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rank-based approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use - random bits on graphs of treewidth at most ; - random bits on planar or -minor free graphs; and - -random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed we obtain an algorithm for detecting a Hamiltonian cycle in an -minor-free graph that runs in deterministic time and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most that use random bits and assign polynomially-bounded weights. We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
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