Robert A. Beeler

Peg solitaire on graphs

There have been several papers on the subject of traditional peg solitaire on different boards. However, in this paper we consider a generalization of the game to arbitrary boards. These boards are treated as graphs in the combinatorial sense. We present necessary and sufficient conditions for the solvability of several well-known families of graphs. In the major result of this paper, we show that the cartesian product of two solvable graphs is likewise solvable. Several related results are also presented. Finally, several open problems related to this study are given.

Double Roman domination

For a graph G = ( V , E ) , a double Roman dominating function is a function f : V ź { 0 , 1 , 2 , 3 } having the property that if f ( v ) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor with f ( w ) = 3 , and if f ( v ) = 1 , then vertex v must have at least one neighbor with f ( w ) ź 2 . The weight of a double Roman dominating function f is the sum f ( V ) = ź v ź V f ( v ) , and the minimum weight of a double Roman dominating function on G is the double Roman domination number of G . We initiate the study of double Roman domination and show its relationship to both domination and Roman domination. Finally, we present an upper bound on the double Roman domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound.

Freely Solvable Graphs in Peg Solitaire

In a 2011 paper, the game of peg solitaire is generalized to arbitrary boards, which are treated as graphs in the combinatorial sense. Of particular interest are graphs that are freely solvable, that is, graphs that can be solved from any starting position. In this paper we give several examples of freely solvable graphs including all such trees with ten vertices or less, numerous cycles with a subdivided chord, meshes, and generalizations of the wheel, helm, and web.

Automorphic Decompositions of Graphs

A decomposition

Curing Instant Insanity II

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Advanced Counting—Inclusion and Exclusion

of a graph H by a graph G is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. The intersection graph

Advanced Counting—Pólya Theory

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