744 research outputs found
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
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
On vertex coloring without monochromatic triangles
We study a certain relaxation of the classic vertex coloring problem, namely,
a coloring of vertices of undirected, simple graphs, such that there are no
monochromatic triangles. We give the first classification of the problem in
terms of classic and parametrized algorithms. Several computational complexity
results are also presented, which improve on the previous results found in the
literature. We propose the new structural parameter for undirected, simple
graphs -- the triangle-free chromatic number . We bound by
other known structural parameters. We also present two classes of graphs with
interesting coloring properties, that play pivotal role in proving useful
observation about our problem. We give/ask several conjectures/questions
throughout this paper to encourage new research in the area of graph coloring.Comment: Extended abstrac
Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus
We extend the notion of canonical ordering (initially developed for planar
triangulations and 3-connected planar maps) to cylindric (essentially simple)
triangulations and more generally to cylindric (essentially internally)
-connected maps. This allows us to extend the incremental straight-line
drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case)
and of Kant (in the -connected case) to this setting. Precisely, for any
cylindric essentially internally -connected map with vertices, we
can obtain in linear time a periodic (in ) straight-line drawing of that
is crossing-free and internally (weakly) convex, on a regular grid
, with and ,
where is the face-distance between the two boundaries. This also yields an
efficient periodic drawing algorithm for graphs on the torus. Precisely, for
any essentially -connected map on the torus (i.e., -connected in the
periodic representation) with vertices, we can compute in linear time a
periodic straight-line drawing of that is crossing-free and (weakly)
convex, on a periodic regular grid
, with and
, where is the face-width of . Since ,
the grid area is .Comment: 37 page
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