Parity dimension for graphs

Combinatorial properties of a general domination problem with parity constraints

“…In this section, we will show that the expected value of the solvability number s(G) is of order 4 n for a random graph in G(n, 1 2 ) (i.e., each edge is inserted with probability 1 2 ), so that the "typical" value of s(G) is pretty close to its maximum. In particular, the following theorem holds:…”

“…The proof of this theorem essentially follows the approach of Amin, Clark and Slater [1]. First of all, fix a vector a ∈ {0, 1} n , and let A denote the (random) adjacency matrix of G. Note that the number of solutions of the matrix equation (A + diag(a))x = 0 over F 2 is exactly X = 2 n−rk(A+diag(a)) .…”

“…Another interesting kind of domination problem involves parity constraints. It has been treated in a series of papers [1][2][3][4]8], motivated by the following remarkable result of Sutner [19]:…”

“….}. It has been treated quite extensively in [1][2][3][4], where the notions of "parity dimension" and "all parity realizable graphs" have been introduced. Wagner [21] gives a recursive procedure for determining the parity dimension of a tree, which is then applied to enumeration problems involving the parity dimension, in particular to counting all parity realizable trees.…”

Connected odd dominating sets in graphs

“…An odd dominating set of a simple, undirected graph G = (V, E) is a set of vertices D ⊆ V such that |N [v] ∩ D| ≡ 1 mod 2 for all vertices v ∈ V , where N [v] denotes the closed neighborhood of v. Odd dominating sets and the analogously defined even dominating sets have received considerable attention in the literature, see [1,3,2,4,5,6,7,8,9,10,11,12,14]. Sutner proved that every graph contains at least one odd dominating set [14] and other proofs of this can be found in [4,8].…”