Proof Techniques and Chessboard Problems Graph Terms informal Independent Set packing anticlique a set of nonadjacent vertices Dominating Set covering a set of vertices that together are adjacent to all the other vertices ID: 766493
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Proof Techniques and Chessboard Problems
Graph Terms (informal) Independent Set : (packing, anti-clique): a set of nonadjacent vertices Dominating Set : (covering): a set of vertices that together are adjacent to all the other vertices. Independent Dominating Set : (non-attacking covering) A set that is both independent and dominating.
More informal terms Open neighborhood of a vertex v, denoted N(v): the set of v’s neighbors Closed neighborhood of a vertex v , denoted N[v], add v to its open neighborhood A set S is maximum with respect to some property P if there are no sets larger than S that have property P. A set S is minimum with respect to some property P if there are no sets smaller than S that have property P.
More definitions A set S is maxim al with respect to some property P if there are no proper supersets of S that have property P . (we cannot add to the S we already have and maintain property P) A set S is minim al with respect to some property P if there are no proper subsets of S that have property P. (we cannot remove any elements from S and maintain property P).
Chromatic Number Given a graph G = (V,E), a proper coloring of the graph assigns colors {1,2,3,…} to the vertices of a graph such that no two adjacent vertices are the same color. Proper coloring: a function f:V N + , such that uv E f(u) f(v). The chromatic number of a graph (G) : the fewest number of colors in a proper coloring.
Grundy Coloring A Grundy coloring is a proper coloring having the property that if a vertex is colored with a color c > 1, then it must be adjacent to at least one vertex with each of the colors 1..c-1. The Grundy number of a graph, denoted (G), is the largest number of colors that can be used in a Grundy coloring of the graph.
Graph Problems Given a graph G = (V,E) For a vertex v Independent Set : a set S V(G) such that u,v S uv E(G) Dominating Set : a set S V(G) such that u S v N(u) S A set S is an independent dominating set if S is both an independent set and a dominating set.
Graph Parameters (G): the size of a minimum dominating set for a graph G. i (G): the size of a minimum independent dominating set for a graph G. (G): the size of a maximum independent set for a graph G. (G): the size of a largest minimal dominating set for a graph G.
Extra graph term Private Neighbor of a vertex v (with respect to S) : Vertex w (it is possible w = v) is a private neighbor of a vertex v S, if N[w] S = {v}. In other words v is the only vertex in S that covers w. A set of vertices is an irredundant set if every vertex in the set has at least one private neighbor. The terms IR(G) and ir (G) refer respectively to the sizes of a largest and a smallest maximal irredundant set in a graph.
Chessboard Graph A graph made from a chessboard, with respect to a type of chess piece. Each square in the chessboard is a vertex. There is an edge between two vertices if a chess piece of interest can move between the two corresponding squares. Varying sizes of chessboards are allowed (other than the 8x8 chessboard). Most work has been done with square chessboards.
Bishops Graph Suppose we are dealing with an nxn chessboard. The bishops graph B n on an nxn chessboard would have: V( B n ) = {1..n} {1..n} E( B n ) = { {(u1, v1), (u2,v2)} | ____________} Help me fill in the blank.