11,657 research outputs found
Edge Dominating Sets and Vertex Covers
Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. These two parameters are used to develop bounds on the vertex cover and total vertex cover numbers of graphs and a resulting chain of vertex covering, edge domination, and matching parameters is explored. In addition, the total vertex cover number is compared to the total domination number of trees and grid graphs
Total domination in inflated graphs
AbstractThe inflation GI of a graph G is obtained from G by replacing every vertex x of degree d(x) by a clique X=Kd(x) and each edge xy by an edge between two vertices of the corresponding cliques X and Y of GI in such a way that the edges of GI which come from the edges of G form a matching of GI. A set S of vertices in a graph G is a total dominating set, abbreviated TDS, of G if every vertex of G is adjacent to a vertex in S. The minimum cardinality of a TDS of G is the total domination number γt(G) of G. In this paper, we investigate total domination in inflated graphs. We provide an upper bound on the total domination number of an inflated graph in terms of its order and matching number. We show that if G is a connected graph of order n≥2, then γt(GI)≥2n/3, and we characterize the graphs achieving equality in this bound. Further, if we restrict the minimum degree of G to be at least 2, then we show that γt(GI)≥n, with equality if and only if G has a perfect matching. If we increase the minimum degree requirement of G to be at least 3, then we show γt(GI)≥n, with equality if and only if every minimum TDS of GI is a perfect total dominating set of GI, where a perfect total dominating set is a TDS with the property that every vertex is adjacent to precisely one vertex of the set
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
k-Tuple_Total_Domination_in_Inflated_Graphs
The inflated graph of a graph with vertices is obtained
from by replacing every vertex of degree of by a clique, which is
isomorph to the complete graph , and each edge of is
replaced by an edge in such a way that , , and
two different edges of are replaced by non-adjacent edges of . For
integer , the -tuple total domination number of is the minimum cardinality of a -tuple total dominating set
of , which is a set of vertices in such that every vertex of is
adjacent to at least vertices in it. For existing this number, must the
minimum degree of is at least . Here, we study the -tuple total
domination number in inflated graphs when . First we prove that
, and then we
characterize graphs that the -tuple total domination number number of
is or . Then we find bounds for this number in the
inflated graph , when has a cut-edge or cut-vertex , in terms
on the -tuple total domination number of the inflated graphs of the
components of or -components of , respectively. Finally, we
calculate this number in the inflated graphs that have obtained by some of the
known graphs
The Disjoint Domination Game
We introduce and study a Maker-Breaker type game in which the issue is to
create or avoid two disjoint dominating sets in graphs without isolated
vertices. We prove that the maker has a winning strategy on all connected
graphs if the game is started by the breaker. This implies the same in the
biased game also in the maker-start game. It remains open to
characterize the maker-win graphs in the maker-start non-biased game, and to
analyze the biased game for . For a more restricted
variant of the non-biased game we prove that the maker can win on every graph
without isolated vertices.Comment: 18 page
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