16 research outputs found
On-line Ramsey numbers of paths and cycles
Consider a game played on the edge set of the infinite clique by two players,
Builder and Painter. In each round, Builder chooses an edge and Painter colours
it red or blue. Builder wins by creating either a red copy of or a blue
copy of for some fixed graphs and . The minimum number of rounds
within which Builder can win, assuming both players play perfectly, is the
on-line Ramsey number . In this paper, we consider the case
where is a path . We prove that for all , and determine
) up to an additive constant for all .
We also prove some general lower bounds for on-line Ramsey numbers of the form
.Comment: Preprin
Total outer-connected domination in trees
Let G = (V,E) be a graph. Set D ⊆ V(G) is a total outer-connected dominating set of G if D is a total dominating set in G and G[V(G)-D] is connected. The total outer-connected domination number of G, denoted by , is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then . Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound
Application of Doubly Connected Dominating Sets to Safe Rectangular Smart Grids
Smart grids, together with the Internet of Things, are considered to be the future of the electric energy world. This is possible through a two-way communication between nodes of the grids and computer processing. It is necessary that the communication is easy and safe, and the distance between a point of demand and supply is short, to reduce the electricity loss. All these requirements should be met at the lowest possible cost. In this paper, we study a two-dimensional rectangular grid graph which is considered to be a model of a smart grid; nodes of the graph represent points and devices of the smart grid, while links represent possible ways of communication and energy transfer. We consider the problem of choosing the lowest possible number of locations (nodes, points) of the grid which could serve as energy sources (or a source of different resources) to other nodes in such a way that we ensure reduction in electricity loss and provide safe communication and resistance to failures and increases in energy demand.Therefore, we study minimum doubly connected dominating sets in grid graphs. We show that the proposed solutions are the best possible in terms of the number of source points for the case of narrow grid graphs and we give upper and lower bounds for the case of wide grid graphs
Application of Doubly Connected Dominating Sets to Safe Rectangular Smart Grids
Smart grids, together with the Internet of Things, are considered to be the future of the electric energy world. This is possible through a two-way communication between nodes of the grids and computer processing. It is necessary that the communication is easy and safe, and the distance between a point of demand and supply is short, to reduce the electricity loss. All these requirements should be met at the lowest possible cost. In this paper, we study a two-dimensional rectangular grid graph which is considered to be a model of a smart grid; nodes of the graph represent points and devices of the smart grid, while links represent possible ways of communication and energy transfer. We consider the problem of choosing the lowest possible number of locations (nodes, points) of the grid which could serve as energy sources (or a source of different resources) to other nodes in such a way that we ensure reduction in electricity loss and provide safe communication and resistance to failures and increases in energy demand.Therefore, we study minimum doubly connected dominating sets in grid graphs. We show that the proposed solutions are the best possible in terms of the number of source points for the case of narrow grid graphs and we give upper and lower bounds for the case of wide grid graphs
A note on on-line Ramsey numbers for quadrilaterals
Tyt. z nagłówka.Bibliogr. s. 468.We consider on-line Ramsey numbers defined by a game played between two players, Builder and Painter. In each round Builder draws an the edge and Painter colors it either red or blue, as it appears. Builder’s goal is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number r(H) of the graph H. An asymmetric version of the on-line Ramsey numbers r(G,H) is defined accordingly. In 2005, Kurek and Ruciński computed r(C3). In this paper, we compute r(C4,Ck) for 3 ≤ k ≤ 7. Most of the results are based on computer algorithms but we obtain the exact value r(C4) and do so without the help of computer algorithms.Dostępny również w formie drukowanej.KEYWORDS: Ramsey theory, on-line games
Graphs with convex domination number close to their order
For a connected graph G = (V,E), a set D ⊆ V(G) is a dominating set of G if every vertex in V(G)-D has at least one neighbour in D. The distance between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length is called an (u-v)-geodesic. A set X ⊆ V(G) is convex in G if vertices from all (a-b)-geodesics belong to X for any two vertices a,b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number of a graph G is the minimum cardinality of a convex dominating set in G. Graphs with the convex domination number close to their order are studied. The convex domination number of a Cartesian product of graphs is also considered
Restricted size Ramsey number for P3 versus cycle
Let F, G and H be simple graphs. We say F → (G,H) if for every 2-coloring of the edges of F there exists a red copy of G or a blue copy of H in F. The Ramsey number r(G,H) is defined as r(G,H) = min{|V(F)|: F → (G,H)}, while the restricted size Ramsey number r*(G,H) is defined as r*(G,H) = min{|E(F)|: F → (G,H),|V(F)| = r(G,H)}. In this paper we determine previously unknown restricted size Ramsey numbers r*(P3,Cn) for 7 ≤ n ≤ 12. We also give new upper bound r*(P3,Cn) ≤ 2n-2 for n ≥ 10 and n is even