88 research outputs found
On the Domination Chain of m by n Chess Graphs
A survey of the six domination chain parameters for both square and rectangular chess boards are discussed
A semi-induced subgraph characterization of upper domination perfect graphs
Let β(G) and Γ(G) be the independence number and the upper domination number of a graph G, respectively. A graph G is called Γ-perfect if β(H) = Γ(H), for every induced subgraph H of G. The class of Γ-perfect graphs generalizes such well-known classes of graphs as strongly perfect graphs, absorbantly perfect graphs, and circular arc graphs. In this article, we present a characterization of Γ-perfect graphs in terms of forbidden semi-induced subgraphs. Key roles in the characterization are played by the odd prism and the even Möbius ladder, where the prism and the Möbius ladder are well-known 3-regular graphs [2]. Using the semi-induced subgraph characterization, we obtain a characterization of K 1.3-free Γ-perfect graphs in terms of forbidden induced subgraphs. © 1999 John Wiley & Sons, Inc
Total irredundance in graphs
AbstractA set S of vertices in a graph G is called a total irredundant set if, for each vertex v in G,v or one of its neighbors has no neighbor in S−{v}. We investigate the minimum and maximum cardinalities of maximal total irredundant sets
Ramsey-type results on parameters related to domination
There is a philosophy to discover Ramsey-type theorem: given a graph
parameter , characterize the family \HH of graphs which satisfies that
every \HH-free graph has bounded parameter . The classical Ramsey's
theorem deals the parameter as the number of vertices. It also has a
corresponding connected version. This Ramsey-type problem on domination number
has been solved by Furuya. We will use this result to handle more parameters
related to domination.Comment: 12 pages, 1 figures
On αrγs(k)-perfect graphs
AbstractFor some integer k⩾0 and two graph parameters π and τ, a graph G is called πτ(k)-perfect, if π(H)−τ(H)⩽k for every induced subgraph H of G. For r⩾1 let αr and γr denote the r-(distance)-independence and r-(distance)-domination number, respectively. In (J. Graph Theory 32 (1999) 303–310), I. Zverovich gave an ingenious complete characterization of α1γ1(k)-perfect graphs in terms of forbidden induced subgraphs. In this paper we study αrγs(k)-perfect graphs for r,s⩾1. We prove several properties of minimal αrγs(k)-imperfect graphs. Generalizing Zverovich's main result in (J. Graph Theory 32 (1999) 303–310), we completely characterize α2r−1γr(k)-perfect graphs for r⩾1. Furthermore, we characterize claw-free α2γ2(k)-perfect graphs
Indicated domination game
Motivated by the success of domination games and by a variation of the
coloring game called the indicated coloring game, we introduce a version of
domination games called the indicated domination game. It is played on an
arbitrary graph by two players, Dominator and Staller, where Dominator
wants to finish the game in as few rounds as possible while Staller wants just
the opposite. In each round, Dominator indicates a vertex of that has
not been dominated by previous selections of Staller, which, by the rules of
the game, forces Staller to select a vertex in the closed neighborhood of .
The game is finished when all vertices of become dominated by the vertices
selected by Staller. Assuming that both players are playing optimally according
to their goals, the number of selected vertices during the game is the
indicated domination number, , of .
We prove several bounds on the indicated domination number expressed in terms
of other graph invariants. In particular, we find a place of the new graph
invariant in the well-known domination chain, by showing that for all graphs , and by showing that the indicated domination
number is incomparable with the game domination number and also with the upper
irredundance number. In connection with the trivial upper bound , we characterize the class of graphs attaining the bound
provided that . We prove that in trees, split graphs and
grids the indicated domination number equals the independence number. We also
find a formula for the indicated domination number of powers of paths, from
which we derive that there exist graphs in which the indicated domination
number is arbitrarily larger than the upper irredundance number.Comment: 19 page
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