14,225 research outputs found
Certified domination
Imagine that we are given a set of officials and a set of civils. For
each civil , there must be an official that can serve ,
and whenever any such is serving , there must also be another civil that observes , that is, may act as a kind of witness, to avoid
any abuse from . What is the minimum number of officials to guarantee such a
service, assuming a given social network?
In this paper, we introduce the concept of certified domination that
perfectly models the aforementioned problem. Specifically, a dominating set
of a graph is said to be certified if every vertex in has
either zero or at least two neighbours in . The cardinality of
a minimum certified dominating set in is called the certified domination
number of . Herein, we present the exact values of the certified domination
number for some classes of graphs as well as provide some upper bounds on this
parameter for arbitrary graphs. We then characterise a wide class of graphs
with equal domination and certified domination numbers and characterise graphs
with large values of certified domination numbers. Next, we examine the effects
on the certified domination number when the graph is modified by
deleting/adding an edge or a vertex. We also provide Nordhaus-Gaddum type
inequalities for the certified domination number. Finally, we show that the
(decision) certified domination problem is NP-complete
Total Dominating Sequences in Graphs
A vertex in a graph totally dominates another vertex if they are adjacent. A
sequence of vertices in a graph is called a total dominating sequence if
every vertex in the sequence totally dominates at least one vertex that was
not totally dominated by any vertex that precedes in the sequence, and at
the end all vertices of are totally dominated. While the length of a
shortest such sequence is the total domination number of , in this paper we
investigate total dominating sequences of maximum length, which we call the
Grundy total domination number, , of . We provide a
characterization of the graphs for which and
of those for which . We show that if is a
nontrivial tree of order~ with no vertex with two or more leaf-neighbors,
then , and characterize the extremal
trees. We also prove that for , if is a connected -regular
graph of order~ different from , then if is not bipartite and
if is
bipartite. The Grundy total domination number is proven to be bounded from
above by two times the Grundy domination number, while the former invariant can
be arbitrarily smaller than the latter. Finally, a natural connection with edge
covering sequences in hypergraphs is established, which in particular yields
the NP-completeness of the decision version of the Grundy total domination
number.Comment: 20 pages, 2 figure
Kernelization and Sparseness: the case of Dominating Set
We prove that for every positive integer and for every graph class
of bounded expansion, the -Dominating Set problem admits a
linear kernel on graphs from . Moreover, when is only
assumed to be nowhere dense, then we give an almost linear kernel on for the classic Dominating Set problem, i.e., for the case . These
results generalize a line of previous research on finding linear kernels for
Dominating Set and -Dominating Set. However, the approach taken in this
work, which is based on the theory of sparse graphs, is radically different and
conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related Connected
Dominating Set problem, the existence of such kernelization algorithms is
unlikely, even though the problem is known to admit a linear kernel on
-topological-minor-free graphs. Also, we prove that for any somewhere dense
class , there is some for which -Dominating Set is
W[]-hard on . Thus, our results fall short of proving a sharp
dichotomy for the parameterized complexity of -Dominating Set on
subgraph-monotone graph classes: we conjecture that the border of tractability
lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded
expansion graph
Semitotal Domination: New hardness results and a polynomial-time algorithm for graphs of bounded mim-width
A semitotal dominating set of a graph with no isolated vertex is a
dominating set of such that every vertex in is within distance two
of another vertex in . The minimum size of a semitotal
dominating set of is squeezed between the domination number and
the total domination number .
\textsc{Semitotal Dominating Set} is the problem of finding, given a graph
, a semitotal dominating set of of size . In this paper,
we continue the systematic study on the computational complexity of this
problem when restricted to special graph classes. In particular, we show that
it is solvable in polynomial time for the class of graphs with bounded
mim-width by a reduction to \textsc{Total Dominating Set} and we provide
several approximation lower bounds for subclasses of subcubic graphs. Moreover,
we obtain complexity dichotomies in monogenic classes for the decision versions
of \textsc{Semitotal Dominating Set} and \textsc{Total Dominating Set}.
Finally, we show that it is -complete to recognise the graphs
such that and those such that , even if restricted to be planar and with maximum degree at
most , and we provide forbidden induced subgraph characterisations for the
graphs heriditarily satisfying either of these two equalities
Total dominator chromatic number of some operations on a graph
Let be a simple graph. A total dominator coloring of is a proper
coloring of the vertices of in which each vertex of the graph is adjacent
to every vertex of some color class. The total dominator chromatic number
of is the minimum number of colors among all total dominator
coloring of . In this paper, we examine the effects on when
is modified by operations on vertex and edge of .Comment: 10 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1511.0165
Locating-dominating sets in twin-free graphs
A locating-dominating set of a graph is a 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-domination number of , denoted
, is the minimum cardinality of a locating-dominating set in .
It is conjectured [D. Garijo, A. Gonz\'alez and A. M\'arquez. The difference
between the metric dimension and the determining number of a graph. Applied
Mathematics and Computation 249 (2014), 487--501] that if is a twin-free
graph of order without isolated vertices, then . We prove the general bound ,
slightly improving over the bound of Garijo et
al. We then provide constructions of graphs reaching the bound,
showing that if the conjecture is true, the family of extremal graphs is a very
rich one. Moreover, we characterize the trees that are extremal for this
bound. We finally prove the conjecture for split graphs and co-bipartite
graphs.Comment: 11 pages; 4 figure
A characterization of trees having a minimum vertex cover which is also a minimum total dominating set
A vertex cover of a graph is a set such that
each edge of is incident to at least one vertex of . A dominating set is a total dominating set of if the subgraph induced by
has no isolated vertices. A -set of is a minimum vertex
cover which is also a minimum total dominating set. In this article we give a
constructive characterization of trees having a -set.Comment: 15 pages, 2, figure
On the roots of total domination polynomial of graphs
Let be a simple graph of order . The total dominating set of
is a subset of that every vertex of is adjacent to some
vertices of . The total domination number of is equal to minimum
cardinality of total dominating set in and denoted by . The
total domination polynomial of is the polynomial
, where is the number of
total dominating sets of of size . In this paper, we study roots of
total domination polynomial of some graphs. We show that all roots of lie in the circle with center and the radius
, where is the minimum degree of . As a
consequence we prove that if , then every integer root
of lies in the set .Comment: 11 pages, 6 figure
Maker-Breaker domination number
The Maker-Breaker domination game is played on a graph by Dominator and
Staller. The players alternatively select a vertex of that was not yet
chosen in the course of the game. Dominator wins if at some point the vertices
he has chosen form a dominating set. Staller wins if Dominator cannot form a
dominating set. In this paper we introduce the Maker-Breaker domination number
of as the minimum number of moves of Dominator to
win the game provided that he has a winning strategy and is the first to play.
If Staller plays first, then the corresponding invariant is denoted
. Comparing the two invariants it turns out that they
behave much differently than the related game domination numbers. The invariant
is also compared with the domination number. Using the
Erd\H{o}s-Selfridge Criterion a large class of graphs is found for which
holds. Residual graphs are introduced and
used to bound/determine and .
Using residual graphs, and are
determined for an arbitrary tree. The invariants are also obtained for cycles
and bounded for union of graphs. A list of open problems and directions for
further investigations is given.Comment: 20 pages, 5 figure
Complexity of the Game Domination Problem
The game domination number is a graph invariant that arises from a game,
which is related to graph domination in a similar way as the game chromatic
number is related to graph coloring. In this paper we show that verifying
whether the game domination number of a graph is bounded by a given integer is
PSPACE-complete. This contrasts the situation of the game coloring problem
whose complexity is still unknown.Comment: 14 pages, 3 figure
- …