276 research outputs found
Two Greedy Consequences for Maximum Induced Matchings
We prove that, for every integer with , there is an
approximation algorithm for the maximum induced matching problem restricted to
-free -regular graphs with performance ratio
, which answers a question posed by Dabrowski et al.
(Theor. Comput. Sci. 478 (2013) 33-40). Furthermore, we show that every graph
with edges that is -degenerate and of maximum degree at most with
, has an induced matching with at least edges
Maximal determinants of combinatorial matrices
We prove that whenever
contains at most ones. We also prove an upper bound on the determinant of
matrices with the -consecutive ones property, a generalisation of the
consecutive ones property, where each row is allowed to have up to blocks
of ones. Finally, we prove an upper bound on the determinant of a path-edge
incidence matrix in a tree and use that to bound the leaf rank of a graph in
terms of its order.Comment: 17 page
Exponential Independence
For a set of vertices of a graph , a vertex in ,
and a vertex in , let be the distance of
and in the graph . Dankelmann et al. (Domination
with exponential decay, Discrete Math. 309 (2009) 5877-5883) define to be
an exponential dominating set of if for every vertex
in , where . Inspired by this
notion, we define to be an exponential independent set of if
for every vertex in , and the
exponential independence number of as the maximum order of an
exponential independent set of .
Similarly as for exponential domination, the non-local nature of exponential
independence leads to many interesting effects and challenges. Our results
comprise exact values for special graphs as well as tight bounds and the
corresponding extremal graphs. Furthermore, we characterize all graphs for
which equals the independence number for every
induced subgraph of , and we give an explicit characterization of all
trees with
Uniquely restricted matchings in subcubic graphs without short cycles
A matching in a graph is uniquely restricted if no other matching in
covers the same set of vertices. We prove that any connected subcubic graph
with vertices and girth at least contains a uniquely restricted
matching of size at least except for two exceptional cubic graphs
of order and
Equality of Distance Packing Numbers
We characterize the graphs for which the independence number equals the
packing number. As a consequence we obtain simple structural descriptions of
the graphs for which (i) the distance--packing number equals the
distance--packing number, and (ii) the distance--matching number equals
the distance--matching number. This last result considerably simplifies and
extends previous results of Cameron and Walker (The graphs with maximum induced
matching and maximum matching the same size, Discrete Math. 299 (2005) 49-55).
For positive integers and with and
, we prove that it is NP-hard to determine
for a given graph whether its distance--packing number equals its
distance--packing number.Comment: 8 page
Reconfiguring dominating sets in minor-closed graph classes
For a graph , two dominating sets and in , and a non-negative
integer , the set is said to -transform to if there is a
sequence of dominating sets in such that ,
, for every , and
arises from by adding or removing one vertex for every . We prove that there is some positive constant and there
are toroidal graphs of arbitrarily large order , and two minimum
dominating sets and in such that -transforms to only
if . Conversely, for every hereditary class
that has balanced separators of order for some
, we prove that there is some positive constant such that, if
is a graph in of order , and and are two dominating sets
in , then -transforms to for
Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees
Let be a graph, and let . For a set of vertices of
, let the set arise by starting with the set , and
iteratively adding further vertices to the current set if they have at
least neighbors in it. If contains all
vertices of , then is known as an irreversible dynamic monopoly or a
perfect target set associated with the threshold function . Let be the minimum cardinality of such an
irreversible dynamic monopoly.
For a connected graph of maximum degree at least , Chang
(Triggering cascades on undirected connected graphs, Information Processing
Letters 111 (2011) 973-978) showed , which was
improved by Chang and Lyuu (Triggering cascades on strongly connected directed
graphs, Theoretical Computer Science 593 (2015) 62-69) to . We show that for every , there is some
such that for every
in , and every connected graph that has maximum
degree at least and girth at least . Furthermore, we show
that for every in , and every tree
that has order at least
Relating broadcast independence and independence
An independent broadcast on a connected graph is a function such that, for every vertex of , the value is at
most the eccentricity of in , and implies that for
every vertex of within distance at most from . The broadcast
independence number of is the largest weight
of an independent broadcast on . Clearly,
is at least the independence number for every
connected graph . Our main result implies . We
prove a tight inequality and characterize all extremal graphs
Approximating Connected Safe Sets in Weighted Trees
For a graph and a non-negative integral weight function on the vertex
set of , a set of vertices of is -safe if for
every component of the subgraph of induced by and every component
of the subgraph of induced by the complement of such that some
vertex in is adjacent to some vertex of . The minimum weight of a
-safe set is the safe number of the weighted graph , and
the minimum weight of a -safe set that induces a connected subgraph of
is its connected safe number . Bapat et al. showed that computing
is NP-hard even when is a star. For a given weighted tree
, they described an efficient -approximation algorithm for
as well as an efficient -approximation algorithm for . Addressing a
problem they posed, we present a PTAS for the connected safe number of a
weighted tree. Our PTAS partly relies on an exact pseudopolynomial time
algorithm, which also allows to derive an asymptotic FPTAS for restricted
instances. Finally, we extend a bound due to Fujita et al. from trees to block
graphs
On some tractable and hard instances for partial incentives and target set selection
A widely studied model for influence diffusion in social networks are {\it
target sets}. For a graph and an integer-valued threshold function
on its vertex set, a {\it target set} or {\it dynamic monopoly} is a set of
vertices of such that iteratively adding to it vertices of that
have at least neighbors in it eventually yields the entire vertex set
of . This notion is limited to the binary choice of including a vertex in
the target set or not, and Cordasco et al.~proposed {\it partial incentives} as
a variant allowing for intermediate choices.
We show that finding optimal partial incentives is hard for chordal graphs
and planar graphs but tractable for graphs of bounded treewidth and for
interval graphs with bounded thresholds. We also contribute some new results
about target set seletion on planar graphs by showing the hardness of this
problem, and by describing an efficient -approximation algorithm
as well as a PTAS for the dual problem of finding a maximum degenerate set
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