8 research outputs found
Bipartite graphs with close domination and k-domination numbers
Let be a positive integer and let be a graph with vertex set .
A subset is a -dominating set if every vertex outside
is adjacent to at least vertices in . The -domination number
is the minimum cardinality of a -dominating set in . For
any graph , we know that where and this bound is sharp for every . In this
paper, we characterize bipartite graphs satisfying the equality for
and present a necessary and sufficient condition for a bipartite graph to
satisfy the equality hereditarily when . We also prove that the problem of
deciding whether a graph satisfies the given equality is NP-hard in general
Domination in graphs of minimum degree at least two and large girth
We prove that for graphs of order n, minimum degree 2 and girth g 5 the domination number satisfies 1 3 + 2 3gn. As a corollary this implies that for cubic graphs of order n and girth g 5 the domination number satisfies 44 135 + 82 135gn which improves recent results due to Kostochka and Stodolsky (An upper bound on the domination number of n-vertex connected cubic graphs, manuscript (2005)) and Kawarabayashi, Plummer and Saito (Domination in a graph with a 2-factor, J. Graph Theory 52 (2006), 1-6) for large enough girth. Furthermore, it confirms a conjecture due to Reed about connected cubic graphs (Paths, stars and the number three, Combin. Prob. Comput. 5 (1996), 267-276) for girth at least 83
In the complement of a dominating set
A set D of vertices of a graph G=(V,E) is a dominating set, if every vertex
of D\V has at least one neighbor that belongs to D. The disjoint domination
number of a graph G is the minimum cardinality of two disjoint dominating
sets of G. We prove upper bounds for the disjoint domination number for
graphs of minimum degree at least 2, for graphs of large minimum degree and
for cubic graphs.A set T of vertices of a graph G=(V,E) is a total
dominating set, if every vertex of G has at least one neighbor that belongs
to T. We characterize graphs of minimum degree 2 without induced 5-cycles
and graphs of minimum degree at least 3 that have a dominating set, a total
dominating set, and a non-empty vertex set that are disjoint.A set I of
vertices of a graph G=(V,E) is an independent set, if all vertices in I are
not adjacent in G. We give a constructive characterization of trees that
have a maximum independent set and a minimum dominating set that are
disjoint and we show that the corresponding decision problem is NP-hard for
general graphs. Additionally, we prove several structural and hardness
results concerning pairs of disjoint sets in graphs which are dominating,
independent, or both. Furthermore, we prove lower bounds for the maximum
cardinality of an independent set of graphs with specifed odd girth and
small average degree.A connected graph G has spanning tree congestion at
most s, if G has a spanning tree T such that for every edge e of T the edge
cut defined in G by the vertex sets of the two components of T-e contains
at most s edges. We prove that every connected graph of order n has
spanning tree congestion at most n^(3/2) and we show that the corresponding
decision problem is NP-hard
Characterization of graphs with equal domination and covering number
AbstractLet G be a simple graph of order n(G). A vertex set D of G is dominating if every vertex not in D is adjacent to some vertex in D, and D is a covering if every edge of G has at least one end in D. The domination number γ(G) is the minimum order of a dominating set, and the covering number β(G) is the minimum order of a covering set in G. In 1981, Laskar and Walikar raised the question of characterizing those connected graphs for which γ(G) = β(G). It is the purpose of this paper to give a complete solution of this problem. This solution shows that the recognition problem, whether a connected graph G has the property γ(G) = β(G), is solvable in polynomial time. As an application of our main results we determine all connected extremal graphs in the well-known inequality γ(G) ⩽ [n(G)2] of Ore (1962), which extends considerable a result of Payan and Xuong from 1982. With a completely different method, independently around the same time, Cockayne, Haynes and Hedetniemi also characterized the connected graphs G with γ(G) = [n(G)2]