8 research outputs found
On the multipacking number of grid graphs
In 2001, Erwin introduced broadcast domination in graphs. It is a variant of
classical domination where selected vertices may have different domination
powers. The minimum cost of a dominating broadcast in a graph is denoted
. The dual of this problem is called multipacking: a multipacking
is a set of vertices such that for any vertex and any positive integer
, the ball of radius around contains at most vertices of .
The maximum size of a multipacking in a graph is denoted mp(G). Naturally
mp(G) . Earlier results by Farber and by Lubiw show that
broadcast and multipacking numbers are equal for strongly chordal graphs. In
this paper, we show that all large grids (height at least 4 and width at least
7), which are far from being chordal, have their broadcast and multipacking
numbers equal
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
2-limited broadcast domination in grid graphs
We establish upper and lower bounds for the 2-limited broadcast domination
number of various grid graphs, in particular the Cartesian product of two
paths, a path and a cycle, and two cycles. The upper bounds are derived by
explicit constructions. The lower bounds are obtained via linear programming
duality by finding lower bounds for the fractional 2-limited multipacking
numbers of these graphs
Broadcasts on Paths and Cycles
A broadcast on a graph is a function such that for every vertex , where denotes the diameter of and the eccentricity of in . The cost of such a broadcast is then the value .Various types of broadcast functions on graphs have been considered in the literature, in relation with domination, irredundence, independenceor packing, leading to the introduction of several broadcast numbers on graphs.In this paper, we determine these broadcast numbers for all paths and cycles, thus answering a questionraised in [D.~Ahmadi, G.H.~Fricke, C.~Schroeder, S.T.~Hedetniemi and R.C.~Laskar, Broadcast irredundance in graphs. {\it Congr. Numer.} 224 (2015), 17--31]
On the multipacking number of grid graphs
In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical domination where selected vertices may have different domination powers. The minimum cost of a dominating broadcast in a graph G is denoted γ_b(G). The dual of this problem is called multipack-ing: a multipacking is a set M ⊆ V(G) such that for any vertex v and any positive integer r, the ball of radius r around v contains at most r vertices of M. The maximum size of a multipacking in a graph G is denoted mp(G). Naturally mp(G) ≤ γ_b(G). Earlier results by Farber and by Lubiw show that broadcast and multipacking numbers are equal for strongly chordal graphs. In this paper, we show that all large grids (height at least 4 and width at least 7), which are far from being chordal, have their broadcast and multipacking numbers equal
Relation Between Broadcast Domination and Multipacking Numbers on Chordal Graphs
International audienceFor a graph G=(V,E) with a vertex set V and an edge set E, a function f:V→{0,1,2,...,diam(G)} is called a broadcast on G. For each vertex u∈V, if there exists a vertex v in G (possibly, u=v) such that f(v)>0 and d(u,v)≤f(v), then f is called a dominating broadcast on G. The cost of the dominating broadcast f is the quantity ∑v∈Vf(v). The minimum cost of a dominating broadcast is the broadcast domination number of G, denoted by γb(G).A multipacking is a set S⊆V in a graph G=(V,E) such that for every vertex v∈V and for every integer r≥1, the ball of radius r around v contains at most r vertices of S, that is, there are at most r vertices in S at a distance at most r from v in G. The multipacking number of G is the maximum cardinality of a multipacking of G and is denoted by mp(G).It is known that mp(G)≤γb(G) and that γb(G)≤2mp(G)+3 for any graph G, and it was shown that γb(G)−mp(G) can be arbitrarily large for connected graphs (as there exist infinitely many connected graphs G where γb(G)/mp(G)=4/3 with mp(G) arbitrarily large). For strongly chordal graphs, it is known that mp(G)=γb(G) always holds.We show that, for any connected chordal graph G, γb(G)≤⌈32mp(G)⌉. We also show that γb(G)−mp(G) can be arbitrarily large for connected chordal graphs by constructing an infinite family of connected chordal graphs such that the ratio γb(G)/mp(G)=10/9, with mp(G) arbitrarily large. This result shows that, for chordal graphs, we cannot improve the bound γb(G)≤⌈32mp(G)⌉ to a bound in the form γb(G)≤c1⋅mp(G)+c2, for any constant c1<10/9 and c2