We consider a variation of cop vs. robber on graph in which the robber is not restricted by the graph edges; instead, he picks a time-independent probability distribution on V (G) and moves according to this fixed distribution. The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously. We show that when the gambler’s distribution...

A graph G is given and two players, a cop and a robber, play the folioking game: the cop chooses a vertex, then the robber chooses a vertex, then the players move alternately beginning with the cop. A move consists of staying at one’s present vertex or moving to an adjacent vertex; each move is seen by both players. The cop wins if he manages to occupy the same vertex as the robber, and the rob...

We investigate generalizations of pebbling numbers and of Graham’s pebbling conjecture that π(G × H) ≤ π(G)π(H), where π(G) is the pebbling number of the graph G. We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that show that Sjöstrand’s theorem on cover p...

Neighbourhood Effects Research at a Crossroads: Ten Challenges for Future Research Neighbourhood effects research is at a crossroads since current theoretical and empirical approaches do not seem to be moving the debate forward. In this paper, we present a set of ten challenges as a basis for a new research agenda which will give new direction to the neighbourhood effects debate. The ten challe...

1 1 Introduction 2 1 Introduction Suppose Γ is a graph with |V (Γ)| = n. For λ a positive integer, let [λ] = {1, 2,. .. , λ} be a set of λ distinct colors. A λ-coloring of Γ is a mapping f from V (Γ) to [λ]. Whenever for every two adjacent vertices u and v, f (u) = f (v), we will call f a proper coloring of Γ; otherwise, improper. When a proper λ-coloring exists, we call Γ a λ-colorable graph. ...

let $g$ be a simple graph of order $n$‎. ‎we consider the‎ ‎independence polynomial and the domination polynomial of a graph‎ ‎$g$‎. ‎the value of a graph polynomial at a specific point can give‎ ‎sometimes a very surprising information about the structure of the‎ ‎graph‎. ‎in this paper we investigate independence and domination‎ ‎polynomial at $-1$ and $1$‎.

Given a polygon P in the plane, a pop operation is the reflection of a vertex with respect to the line through its adjacent vertices. We define a family of alternating polygons, and show that any polygon from this family cannot be convexified by pop operations. This family contains simple, as well as non-simple (i.e., self-intersecting) polygons, as desired. We thereby answer in the negative an...

The pebbling number f(G) of a graphG is the least p such that, no matter how p pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H , we have f(G × H) ≤ f(G)f(H). If the graph G satisfies the odd 2-pebbling property, we w...

APrIl 2017 IZA DP No. 10694 Incorporating Neighbourhood Choice in a Model of Neighbourhood Effects on Income Studies of neighbourhood effects often attempt to identify causal effects of neighbourhood characteristics on individual outcomes, such as income, education, employment, and health. However, selection looms large in this line of research and it has been repeatedly argued that estimates o...