Broadcast independence number of oriented circulant graphs
CoRR（2024）
摘要
In 2001, D. Erwin introduced in his Ph.D. dissertation the
notion of broadcast independence in unoriented graphs. Since then, some results
but not many, are published on this notion, including research work on the
broadcast independence number of unoriented circulant graphs . In
this paper, we are focused in the same parameter but of the class of oriented
circulant graphs. An independent broadcast on an oriented graph
G is a function f: V⟶{0,…,(G)} such that (i) f(v)≤ e(v) for
every vertex v∈ V(G), where (G)
denotes the diameter of G and e(v) the eccentricity of
vertex v, and (ii) d_G(u,v) > f(u) for every distinct
vertices u, v with f(u), f(v)>0, where d_G(u,v)
denotes the length of a shortest oriented path from u to v. The broadcast
independence number β_b(G) of G is
then the maximum value of ∑_v ∈ V f(v), taken over all independent
broadcasts on G. The goal of this paper is to study the
properties of independent broadcasts of oriented circulant graphs
C(n;1,a), for any integers n and a with n>|a|≥ 1
and a ∉{1,n-1}. Then, we give some bounds and some exact values for
the number β_b(C(n;1,a)).
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