23,074 research outputs found
On bounding the difference between the maximum degree and the chromatic number by a constant
We provide a finite forbidden induced subgraph characterization for the graph
class , for all , which is defined as
follows. A graph is in if for any induced subgraph, holds, where is the maximum degree and is the
chromatic number of the subgraph.
We compare these results with those given in [O. Schaudt, V. Weil, On
bounding the difference between the maximum degree and the clique number,
Graphs and Combinatorics 31(5), 1689-1702 (2015). DOI:
10.1007/s00373-014-1468-3], where we studied the graph class , for
, whose graphs are such that for any induced subgraph,
holds, where denotes the clique number of
a graph. In particular, we give a characterization in terms of
and of those graphs where the neighborhood of every vertex is
perfect.Comment: 10 pages, 4 figure
Optimal accessing and non-accessing structures for graph protocols
An accessing set in a graph is a subset B of vertices such that there exists
D subset of B, such that each vertex of V\B has an even number of neighbors in
D. In this paper, we introduce new bounds on the minimal size kappa'(G) of an
accessing set, and on the maximal size kappa(G) of a non-accessing set of a
graph G. We show strong connections with perfect codes and give explicitly
kappa(G) and kappa'(G) for several families of graphs. Finally, we show that
the corresponding decision problems are NP-Complete
The leafage of a chordal graph
The leafage l(G) of a chordal graph G is the minimum number of leaves of a
tree in which G has an intersection representation by subtrees. We obtain upper
and lower bounds on l(G) and compute it on special classes. The maximum of l(G)
on n-vertex graphs is n - lg n - (1/2) lg lg n + O(1). The proper leafage l*(G)
is the minimum number of leaves when no subtree may contain another; we obtain
upper and lower bounds on l*(G). Leafage equals proper leafage on claw-free
chordal graphs. We use asteroidal sets and structural properties of chordal
graphs.Comment: 19 pages, 3 figure
On Weak Odd Domination and Graph-based Quantum Secret Sharing
A weak odd dominated (WOD) set in a graph is a subset B of vertices for which
there exists a distinct set of vertices C such that every vertex in B has an
odd number of neighbors in C. We point out the connections of weak odd
domination with odd domination, [sigma,rho]-domination, and perfect codes. We
introduce bounds on \kappa(G), the maximum size of WOD sets of a graph G, and
on \kappa'(G), the minimum size of non WOD sets of G. Moreover, we prove that
the corresponding decision problems are NP-complete. The study of weak odd
domination is mainly motivated by the design of graph-based quantum secret
sharing protocols: a graph G of order n corresponds to a secret sharing
protocol which threshold is \kappa_Q(G) = max(\kappa(G), n-\kappa'(G)). These
graph-based protocols are very promising in terms of physical implementation,
however all such graph-based protocols studied in the literature have
quasi-unanimity thresholds (i.e. \kappa_Q(G)=n-o(n) where n is the order of the
graph G underlying the protocol). In this paper, we show using probabilistic
methods, the existence of graphs with smaller \kappa_Q (i.e. \kappa_Q(G)<
0.811n where n is the order of G). We also prove that deciding for a given
graph G whether \kappa_Q(G)< k is NP-complete, which means that one cannot
efficiently double check that a graph randomly generated has actually a
\kappa_Q smaller than 0.811n.Comment: Subsumes arXiv:1109.6181: Optimal accessing and non-accessing
structures for graph protocol
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
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