Combinatorial properties of a general domination problem with parity constraints

Abstract

We consider various properties of a general parity domination problem: given a graph G on n vertices, one is looking for a subset S of the vertex set such that the open/closed neighborhood of each vertex contains an even/odd number of vertices in S (it is prescribed individually for each vertex which of these applies). We define the parameter s(G) to be the number of solvable instances out of 4 n possibilities and study the properties of this parameter. Upper and lower bounds for general graphs and trees are given as well as a remarkable recurrence formula for rooted trees. Furthermore, we give explicit formulas in several special cases and investigate random graphs.

Extremal Trees with Fixed Degree Sequence

“…It was shown in [15] that the path has the greatest solvability among all trees (in fact all graphs) of a given order, while the star has the least solvability, but no other extremal results for the solvability of trees are available. Let us show that our general results apply to the solvability as well.…”

“…The only tree with this degree sequence is the star S n = G(B) = M(B). Therefore, we obtain the following known results as immediate corollaries: among all n-vertex trees, the star minimises the Wiener index [9], the Hosoya index and the energy [11], the quantity rf(T, x) [43] for every x > 0 (thus also the incidence energy) and the solvability [15], while it maximises the number of subtrees [30] and the Merrifield-Simmons index [28].…”

“…G be a graph and v a vertex in G. The open neighbourhood of v, denoted N (v), is the set {u ∈ V (G) : uv ∈ E(G)}, and the closed neighbourhood ofv is N [v] = N (v) ∪ {v}.Let F 2 be the field with two elements. The solvability of G, introduced in[15] and denoted s(G), is the number of pairs (a, b) ∈ F…”