459,141 research outputs found
Edge Roman domination on graphs
An edge Roman dominating function of a graph is a function satisfying the condition that every edge with
is adjacent to some edge with . The edge Roman
domination number of , denoted by , is the minimum weight
of an edge Roman dominating function of .
This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad stating that if is a graph of maximum degree
on vertices, then . While the counterexamples having the edge Roman domination numbers
, we prove that is an upper bound for connected graphs. Furthermore, we
provide an upper bound for the edge Roman domination number of -degenerate
graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic
graphs.
In addition, we prove that the edge Roman domination numbers of planar graphs
on vertices is at most , which confirms a conjecture of
Akbari and Qajar. We also show an upper bound for graphs of girth at least five
that is 2-cell embeddable in surfaces of small genus. Finally, we prove an
upper bound for graphs that do not contain as a subdivision, which
generalizes a result of Akbari and Qajar on outerplanar graphs
A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs
We present a near-optimal polynomial-time approximation algorithm for the
asymmetric traveling salesman problem for graphs of bounded orientable or
non-orientable genus. Our algorithm achieves an approximation factor of O(f(g))
on graphs with genus g, where f(n) is the best approximation factor achievable
in polynomial time on arbitrary n-vertex graphs. In particular, the
O(log(n)/loglog(n))-approximation algorithm for general graphs by Asadpour et
al. [SODA 2010] immediately implies an O(log(g)/loglog(g))-approximation
algorithm for genus-g graphs. Our result improves the
O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi [SODA
2011], which applies only to graphs with orientable genus g; ours is the first
approximation algorithm for graphs with bounded non-orientable genus.
Moreover, using recent progress on approximating the genus of a graph, our
O(log(g) / loglog(g))-approximation can be implemented even without an
embedding when the input graph has bounded degree. In contrast, the
O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi requires a
genus-g embedding as part of the input.
Finally, our techniques lead to a O(1)-approximation algorithm for ATSP on
graphs of genus g, with running time 2^O(g)*n^O(1)
Further Studies on the Sparing Number of Graphs
Let denote the set of all non-negative integers and
be its power set. An integer additive set-indexer
is an injective function such that the
induced function defined by is also injective, where is the sum set of and
. If , then is said to be a -uniform
integer additive set-indexer. An integer additive set-indexer is said to be
a weak integer additive set-indexer if . In this paper, we study the admissibility of weak integer additive
set-indexer by certain graphs and graph operations.Comment: 10 Pages, Submitted. arXiv admin note: substantial text overlap with
arXiv:1310.609
A Study on Integer Additive Set-Graceful Graphs
A set-labeling of a graph is an injective function , where is a finite set and a set-indexer of is a
set-labeling such that the induced function defined by
for every is also injective. An integer additive set-labeling is
an injective function ,
is the set of all non-negative integers and an integer additive
set-indexer is an integer additive set-labeling such that the induced function
defined by is also injective. In this paper, we extend the concepts of set-graceful
labeling to integer additive set-labelings of graphs and provide some results
on them.Comment: 11 pages, submitted to JARP
On Integer Additive Set-Indexers of Graphs
A set-indexer of a graph is an injective set-valued function such that the function
defined by for every is also injective, where is
the set of all subsets of and is the symmetric difference of sets.
An integer additive set-indexer is defined as an injective function
such that the induced function defined by is also
injective. A graph which admits an IASI is called an IASI graph. An IASI
is said to be a {\em weak IASI} if and an
IASI is said to be a {\em strong IASI} if for all
. In this paper, we study about certain characteristics of inter
additive set-indexers.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1312.7674 To
Appear in Int. J. Math. Sci.& Engg. Appl. in March 201
A Study on Integer Additive Set-Valuations of Signed Graphs
Let denote the set of all non-negative integers and \cP(\N) be its
power set. An integer additive set-labeling (IASL) of a graph is an
injective set-valued function f:V(G)\to \cP(\N)-\{\emptyset\} such that the
induced function f^+:E(G) \to \cP(\N)-\{\emptyset\} is defined by , where is the sumset of and . A graph
which admits an IASL is usually called an IASL-graph. An IASL of a graph
is said to be an integer additive set-indexer (IASI) of if the
associated function is also injective. In this paper, we define the
notion of integer additive set-labeling of signed graphs and discuss certain
properties of signed graphs which admits certain types of integer additive
set-labelings.Comment: 12 pages, Carpathian Mathematical Publications, Vol. 8, Issue 2,
2015, 12 page
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