The process succeeds, by reducing the graph to a single vertex, if and only if the graph is cop-win. It is even unknown whether the soft Meyniel conjecture, that there exists a constant c < 1 {\displaystyle c<1} for which the cop number is always O ( n c ) {\displaystyle O(n The hereditarily cop-win graphs are the same as the bridged graphs, graphs in which every cycle of length four or more has a shortcut, a pair of vertices closer in the graph than they are in the cycle. Conversely, almost all dismantlable graphs have a universal vertex, in the sense that, among all n-vertex dismantlable graphs, the fraction of these graphs that have a universal vertex goes to one in the limit as n goes to infinity.

The cop can win in a strong product of two cop-win graphs by, first, playing to win in one of these two factor graphs, reaching a pair whose first component is the same as the robber. Additionally, if v is a dominated vertex in a cop-win graph, then removing v must produce another cop-win graph, for otherwise the robber could play within that subgraph, pretending that the cop is on the vertex that dominates v whenever the cop is actually on v, and never get caught. Gavenčiak, Tomáš (2010), "Cop-win graphs with maximum capture-time", Discrete Mathematics, 310 (10–11): 1557–1563, doi: 10. Lubiw, Anna; Snoeyink, Jack; Vosoughpour, Hamideh (2017), "Visibility graphs, dismantlability, and the cops and robbers game", Computational Geometry, 66: 14–27, arXiv: 1601. These are graphs defined from the vertices of a polygon, with an edge whenever two vertices can be connected by a line segment that does not pass outside the polygon.By the induction hypothesis, the cop has a winning strategy on the graph formed by removing v, and can follow the same strategy on the original graph by pretending that the robber is on the vertex that dominates v whenever the robber is actually on v.

Make sure everyone understands what contact is acceptable, and monitor contact throughout the activity. Construct a block of the log n removed vertices and numbers representing all other vertices' adjacencies within this block. A family of mathematical objects is said to be closed under a set of operations if combining members of the family always produces another member of that family. However, there exist infinite chordal graphs, and even infinite chordal graphs of diameter two, that are not cop-win.On each subsequent turn, the player controlling the cops chooses a (possibly empty) subset of the cops, and moves each of these cops to adjacent vertices. Repeatedly find a vertex v that is an endpoint of an edge participating in a number of triangles equal to the degree of v minus one, delete v, and decrement the triangles per edge of each remaining edge that formed a triangle with v. Construct the deficit set for all adjacent pairs that have deficit at most log n and that have not already had this set constructed.

Henri Meyniel (also known for Meyniel graphs) conjectured in 1985 that every connected n {\displaystyle n} -vertex graph has cop number O ( n ) {\displaystyle O({\sqrt {n}})} . a b c d e f g h i Nowakowski, Richard; Winkler, Peter (1983), "Vertex-to-vertex pursuit in a graph", Discrete Mathematics, 43 (2–3): 235–239, doi: 10.If this number becomes zero, after other vertices have been removed, then x is dominated by y and may also be removed.