285 research outputs found
A note on acyclic domination number in graphs of diameter two
Author name used in this publication: C. T. Ng2005-2006 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe
The Minimum Shared Edges Problem on Grid-like Graphs
We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide
whether it is possible to route paths from a start vertex to a target
vertex in a given graph while using at most edges more than once. We show
that MSE can be decided on bounded (i.e. finite) grids in linear time when both
dimensions are either small or large compared to the number of paths. On
the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids.
Finally, we study MSE from a parametrised complexity point of view. It is known
that MSE is fixed-parameter tractable with respect to the number of paths.
We show that, under standard complexity-theoretical assumptions, the problem
parametrised by the combined parameter , , maximum degree, diameter, and
treewidth does not admit a polynomial-size problem kernel, even when restricted
to planar graphs
Locating-total dominating sets in twin-free graphs: a conjecture
A total dominating set of a graph is a set of vertices of such
that every vertex of has a neighbor in . A locating-total dominating set
of is a total dominating set of with the additional property that
every two distinct vertices outside have distinct neighbors in ; that
is, for distinct vertices and outside , where denotes the open neighborhood of . A graph is twin-free if
every two distinct vertices have distinct open and closed neighborhoods. The
location-total domination number of , denoted , is the minimum
cardinality of a locating-total dominating set in . It is well-known that
every connected graph of order has a total dominating set of size at
most . We conjecture that if is a twin-free graph of order
with no isolated vertex, then . We prove the
conjecture for graphs without -cycles as a subgraph. We also prove that if
is a twin-free graph of order , then .Comment: 18 pages, 1 figur
Global Domination Stable Graphs
A set of vertices S in a graph G is a global dominating set (GDS) of G if S is a dominating set for both G and its complement G. The minimum cardinality of a global dominating set of G is the global domination number of G. We explore the effects of graph modifications on the global domination number. In particular, we explore edge removal, edge addition, and vertex removal
Parameterized Inapproximability of Target Set Selection and Generalizations
In this paper, we consider the Target Set Selection problem: given a graph
and a threshold value for any vertex of the graph, find a minimum
size vertex-subset to "activate" s.t. all the vertices of the graph are
activated at the end of the propagation process. A vertex is activated
during the propagation process if at least of its neighbors are
activated. This problem models several practical issues like faults in
distributed networks or word-to-mouth recommendations in social networks. We
show that for any functions and this problem cannot be approximated
within a factor of in time, unless FPT = W[P],
even for restricted thresholds (namely constant and majority thresholds). We
also study the cardinality constraint maximization and minimization versions of
the problem for which we prove similar hardness results
Location-domination in line graphs
A set of vertices of a graph is locating if every two distinct
vertices outside have distinct neighbors in ; that is, for distinct
vertices and outside , , where
denotes the open neighborhood of . If is also a dominating set (total
dominating set), it is called a locating-dominating set (respectively,
locating-total dominating set) of . A graph is twin-free if every two
distinct vertices of have distinct open and closed neighborhoods. It is
conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the
metric dimension and the determining number of a graph. Applied Mathematics and
Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning.
Locating-total dominating sets in twin-free graphs: a conjecture. The
Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any
twin-free graph without isolated vertices has a locating-dominating set of
size at most one-half its order and a locating-total dominating set of size at
most two-thirds its order. In this paper, we prove these two conjectures for
the class of line graphs. Both bounds are tight for this class, in the sense
that there are infinitely many connected line graphs for which equality holds
in the bounds.Comment: 23 pages, 2 figure
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