58 research outputs found
The Optimal Rubbling Number of Ladders, Prisms and M\"obius-ladders
A pebbling move on a graph removes two pebbles at a vertex and adds one
pebble at an adjacent vertex. Rubbling is a version of pebbling where an
additional move is allowed. In this new move, one pebble each is removed at
vertices and adjacent to a vertex , and an extra pebble is added at
vertex . A vertex is reachable from a pebble distribution if it is possible
to move a pebble to that vertex using rubbling moves. The optimal rubbling
number is the smallest number needed to guarantee a pebble distribution of
pebbles from which any vertex is reachable. We determine the optimal
rubbling number of ladders (), prisms () and
M\"oblus-ladders
The complexity of recognizing minimally tough graphs
A graph is called -tough if the removal of any vertex set that
disconnects the graph leaves at most components. The toughness of a
graph is the largest for which the graph is -tough. A graph is minimally
-tough if the toughness of the graph is and the deletion of any edge
from the graph decreases the toughness. The complexity class DP is the set of
all languages that can be expressed as the intersection of a language in NP and
a language in coNP. In this paper, we prove that recognizing minimally
-tough graphs is DP-complete for any positive rational number . We
introduce a new notion called weighted toughness, which has a key role in our
proof
Properties of minimally -tough graphs
A graph is minimally -tough if the toughness of is and the
deletion of any edge from decreases the toughness. Kriesell conjectured
that for every minimally -tough graph the minimum degree . We
show that in every minimally -tough graph . We
also prove that every minimally -tough claw-free graph is a cycle. On the
other hand, we show that for every any graph can be embedded
as an induced subgraph into a minimally -tough graph
Constructions for the optimal pebbling of grids
In [C. Xue, C. Yerger: Optimal Pebbling on Grids, Graphs and Combinatorics]
the authors conjecture that if every vertex of an infinite square grid is
reachable from a pebble distribution, then the covering ratio of this
distribution is at most . First we present such a distribution with
covering ratio , disproving the conjecture. The authors in the above paper
also claim to prove that the covering ratio of any pebble distribution is at
most . The proof contains some errors. We present a few interesting
pebble distributions that this proof does not seem to cover and highlight some
other difficulties of this topic
SIS epidemic propagation on hypergraphs
Mathematical modeling of epidemic propagation on networks is extended to
hypergraphs in order to account for both the community structure and the
nonlinear dependence of the infection pressure on the number of infected
neighbours. The exact master equations of the propagation process are derived
for an arbitrary hypergraph given by its incidence matrix. Based on these,
moment closure approximation and mean-field models are introduced and compared
to individual-based stochastic simulations. The simulation algorithm, developed
for networks, is extended to hypergraphs. The effects of hypergraph structure
and the model parameters are investigated via individual-based simulation
results
Minimally toughness in special graph classes
Let be a positive real number. A graph is called -tough, if the
removal of any cutset leaves at most components. The toughness of a
graph is the largest for which the graph is -tough. A graph is minimally
-tough, if the toughness of the graph is and the deletion of any edge
from the graph decreases the toughness. In this paper we investigate the
minimum degree and the recognizability of minimally -tough graphs in the
class of chordal graphs, split graphs, claw-free graphs and -free graphs
Upper Bound on the Optimal Rubbling Number in graphs with given minimum degree
A pebbling move on a graph removes two pebbles at a vertex and adds
one pebble at an adjacent vertex. A vertex is reachable
from a pebble distribution if it is possible to move a pebble to
that vertex using pebbling moves. The optimal pebbling number is
the smallest number needed to guarantee a pebble distribution of
pebbles from which any vertex is reachable. Czygrinow proved that
the optimal pebbling number of a graph is at most , where is the number of the vertices and is
the minimum degree of the graph. We improve this result and show that the optimal pebbling number is at most
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