5,854 research outputs found
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines
In the paper we consider the problem of scheduling identical jobs on 4
uniform machines with speeds respectively.
Our aim is to find a schedule with a minimum possible length. We assume that
jobs are subject to some kind of mutual exclusion constraints modeled by a
bipartite incompatibility graph of degree , where two incompatible jobs
cannot be processed on the same machine. We show that the problem is NP-hard
even if . If, however, and ,
, then the problem can be solved to optimality in time
. The same algorithm returns a solution of value at most 2 times
optimal provided that . Finally, we study the case and give an -time -approximation algorithm in
all such situations
Generalized Distance Domination Problems and Their Complexity on Graphs of Bounded mim-width
We generalize the family of (sigma, rho)-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-r dominating set and distance-r independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and quickly computable, such as k-trapezoid graphs, Dilworth k-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, k-polygon graphs, circular arc graphs, complements of d-degenerate graphs, and H-graphs if given an H-representation. To supplement these findings, we show that many classes of (distance) (sigma, rho)-problems are W[1]-hard parameterized by mim-width + solution size
Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles
Given a simple graph , a subset of is called a triangle cover if
it intersects each triangle of . Let and denote the
maximum number of pairwise edge-disjoint triangles in and the minimum
cardinality of a triangle cover of , respectively. Tuza conjectured in 1981
that holds for every graph . In this paper, using a
hypergraph approach, we design polynomial-time combinatorial algorithms for
finding small triangle covers. These algorithms imply new sufficient conditions
for Tuza's conjecture on covering and packing triangles. More precisely,
suppose that the set of triangles covers all edges in . We
show that a triangle cover of with cardinality at most can be
found in polynomial time if one of the following conditions is satisfied: (i)
, (ii) , (iii)
.
Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs,
Combinatorial algorithm
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