331 research outputs found
Polynomial kernelization for removing induced claws and diamonds
A graph is called (claw,diamond)-free if it contains neither a claw (a
) nor a diamond (a with an edge removed) as an induced subgraph.
Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of
triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex
is in at most two maximal cliques and every edge is in exactly one maximal
clique.
In this paper we consider the parameterized complexity of the
(claw,diamond)-free Edge Deletion problem, where given a graph and a
parameter , the question is whether one can remove at most edges from
to obtain a (claw,diamond)-free graph. Our main result is that this problem
admits a polynomial kernel. We complement this finding by proving that, even on
instances with maximum degree , the problem is NP-complete and cannot be
solved in time unless the Exponential Time
Hypothesis fai
The Cycle Spectrum of Claw-free Hamiltonian Graphs
If is a claw-free hamiltonian graph of order and maximum degree
with , then has cycles of at least many different lengths.Comment: 9 page
Packing 3-vertex paths in claw-free graphs and related topics
An L-factor of a graph G is a spanning subgraph of G whose every component is
a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the
domination number of G. A claw is a graph with four vertices and three edges
incident to the same vertex. A graph is claw-free if it has no induced subgraph
isomorphic to a claw. Our results include the following. Let G be a 3-connected
claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in
G. Then
(a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2)
if v(G) = 1 mod 3, then G - x has an L-factor, (a3) if v(G) = 2 mod 3, then G -
{x,y} has an L-factor, (a4) if v(G) = 0 mod 3 and G is either cubic or
4-connected, then G - P has an L-factor, (a5) if G is cubic with v(G) > 5 and E
is a set of three edges in G, then G - E has an L-factor if and only if the
subgraph induced by E in G is not a claw and not a triangle, (a6) if v(G) = 1
mod 3, then G - {v,e} has an L-factor for every vertex v and every edge e in G,
(a7) if v(G) = 1 mod 3, then there exist a 4-vertex path N and a claw Y in G
such that G - N and G - Y have L-factors, and (a8) d(G) < v(G)/3 +1 and if in
addition G is not a cycle and v(G) = 1 mod 3, then d(G) < v(G)/3.
We explore the relations between packing problems of a graph and its line
graph to obtain some results on different types of packings. We also discuss
relations between L-packing and domination problems as well as between induced
L-packings and the Hadwiger conjecture.
Keywords: claw-free graph, cubic graph, vertex disjoint packing, edge
disjoint packing, 3-vertex factor, 3-vertex packing, path-factor, induced
packing, graph domination, graph minor, the Hadwiger conjecture.Comment: 29 page
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