Threat, support and dead edges in the Shannon game

Abstract

The notions of captured/lost vertices and dead edges in the Shannon game (Shannon switching game on nodes) are examined using graph theory. Simple methods are presented for identifying some dead edges and some captured sets of vertices, thus simplifying the (computationally hard) problem of analyzing the game.

Minimal and irreducible links in the Shannon game

“…The notation follows that of [4]. The Shannon game (G, s, t) is played on a simple graph G (no self-loops, no multiple edges) with two vertices s, t designated as terminals.…”

“…We will make use of several results from 'Threat, support and dead edges in the Shannon game' [4]. The notation 'theorem TSD.n' refers to theorem n from that paper.…”

“…Suppose a, b are mutually surrounding and so are a, c. Then c is redundant, that is, c and its edges can be deleted without changing the outcome of the game. This is because a mutually surrounding pair such as a, b is 'captured' [4], meaning both can be filled in by Short as a free move, without changing the outcome of the game. After that filling in, c becomes simplicial (its neighbours form a clique) so it is dead (see [1] and theorem TSD.13).…”

“…Having identified candidate terminals, one can look among the other vertices in the sure knowledge that none will be assigned as a terminal. If any of those are mutually threatening then they are 'lost' [4], that is, they can be deleted without changing the outcome of the game. This would imply the link was not minimal.…”

“…In the absence of an efficient algorithm for this general task, the most natural strategy is to try to break down the task into sub-games. This leads to the concept of 'strong' and 'weak' virtual connections or links, and to the concept of the multi-Shannon game, see [1,3,4]. A 'link' (we prefer this term to 'virtual connection') is a set of vertices and edges which suffices to allow one player (Short) to guarantee to form a path between two vertices at the ends of the link, following the rules of the game.…”