The edge-flipping group of a graph

Abstract

Let X = (V, E) be a finite simple connected graph with n vertices and m edges. A configuration is an assignment of one of two colors, black or white, to each edge of X. A move applied to a configuration is to select a black edge ǫ ∈ E and change the colors of all adjacent edges of ǫ. Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping puzzle on X, and it corresponds to a group action. This group is called the edge-flipping group W E (X) of X. This paper shows that if X has at least three vertices, W E (X) is isomorphic to a semidirect product of (Z/2Z) k and the symmetric group S n of degree n, where k = (n − 1)(m − n + 1) if n is odd, k = (n − 2)(m − n + 1) if n is even, and Z is the additive group of integers.An ordered pair X = (V, E) is a finite simple graph if V is a finite set and E is a set of some 2-element subsets of V. The elements of V are called vertices of X and the elements of E are called edges of X. Two vertices

Lit-only σ-game on pseudo-trees

“…Finally, as hinted by Eqs. (1) and (2), the lit-only σ -game is naturally associated with certain groups generated by transvections and hence is also studied (implicitly) in some algebra settings [8,9].…”

Minimum Light Numbers in the $\sigma$-Game and Lit-Only $\sigma$-Game on Unicyclic and Grid Graphs

“…The lit-only σ-game and its closely related variants have been studied not only for fun by amateurs [40] but are also studied by mathematicians for mathematical fun [8,10,25,26,44,45,46,48] and from the perspectives of error-correcting codes and combinatorial game theory [12,13,14,15,17], Lie algebras and Coxeter groups [4,5,6,7,29,30,31,32,39], statistical physics of social balance [33,34], and general reachability analysis [27]. The study of the σ-game has a longer history than that of the lit-only σ-game and is still mushrooming; see [1,2,3,9,10,11,16,18,19,20,21,22,23,24,25,28,35,36,40,41,42,…”

Lit-only sigma-game on nondegenerate graphs

“…Moreover, if G is a tree of order n ≥ 3, Wu showed that H Γ is isomorphic to the symmetric group on n letters. Weng and the author [12] determined the structure of H Γ without any assumption on G. The lit-only σ-game on a simple graph Γ can also be considered as a representation κ Γ of the simply-laced Coxeter group W Γ over F 2 [11]. The dual representation of κ Γ preserves a certain symplectic form B Γ .…”

“…Moreover we show that these graphs Γ are 2-lit and provide a linear algebraic criterion for Γ to be 1-lit (Theorem 3.2). Combining Theorem 3.1, Theorem 3.2 and those in [12] and [19], the study of the lit-only σ-game on nondegenerate graphs is quite completed and the focus for further research is on degenerate graphs.…”