486 research outputs found
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
Approximation algorithm for finding multipacking on Cactus
For a graph with vertex set and edge set , a
function is called a
\emph{broadcast} on . For each vertex , if there exists a
vertex in (possibly, ) such that and , then is called a \textit{dominating broadcast} on .
The \textit{cost} of the dominating broadcast is the quantity . The minimum cost of a dominating broadcast is the \textit{broadcast
domination number} of , denoted by . A
\textit{multipacking} is a set in a graph such
that for every vertex and for every integer , the ball
of radius around contains at most vertices of , that is,
there are at most vertices in at a distance at most from in . The \textit{multipacking number} of is the maximum
cardinality of a multipacking of and is denoted by . We show
that, for any cactus graph , . We also show that can be
arbitrarily large for cactus graphs by constructing an infinite family of
cactus graphs such that the ratio , with
arbitrarily large. This result shows that, for cactus graphs, we cannot improve
the bound to a bound in the
form , for any constant and
. Moreover, we provide an -time algorithm to construct a
multipacking of of size at least , where
is the number of vertices of the graph
Finding Optimal 2-Packing Sets on Arbitrary Graphs at Scale
A 2-packing set for an undirected graph is a subset such that any two vertices have no common
neighbors. Finding a 2-packing set of maximum cardinality is a NP-hard problem.
We develop a new approach to solve this problem on arbitrary graphs using its
close relation to the independent set problem. Thereby, our algorithm red2pack
uses new data reduction rules specific to the 2-packing set problem as well as
a graph transformation. Our experiments show that we outperform the
state-of-the-art for arbitrary graphs with respect to solution quality and also
are able to compute solutions multiple orders of magnitude faster than
previously possible. For example, we are able to solve 63% of our graphs to
optimality in less than a second while the competitor for arbitrary graphs can
only solve 5% of the graphs in the data set to optimality even with a 10 hour
time limit. Moreover, our approach can solve a wide range of large instances
that have previously been unsolved
On Split-Coloring Problems
We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this are
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