Temel Kavramlar
The author establishes a lower bound for the secure domination number of outerplanar graphs, proving that it is at least (n + 4)/5 and demonstrating its tightness.
Özet
In this paper, the author delves into the concept of secure dominating sets in outerplanar graphs. The study focuses on proving a lower bound for the secure domination number, showcasing that for any outerplanar graph with n vertices greater than or equal to 4, the secure domination number is at least (n + 4)/5. The research highlights the significance of this lower bound and provides insights into the structure and properties of secure dominating sets in outerplanar graphs. Through rigorous mathematical proofs and analysis, the study contributes to the understanding of secure domination in graph theory.
İstatistikler
For any outerplanar graph with n ≥ 4 vertices, γs(G) ≥ (n + 4)/5.
The author proved that γs(G) ≤ ⌈3n/7⌉ for an outerplanar graph.
In stripped maximal outerplanar graphs, n/4 < γs(G) ≤ ⌈n/3⌉.
A graph G is outerplanar if it has a crossing-free embedding in the plane such that all vertices belong to the boundary of its outer face.
Campos and Wakabayashi showed that if G is a maximal outerplanar graph with n vertices, then γ(G) ≤ (n + k)/4 where k is the number of vertices of degree 2.
Alıntılar
"The problem of secure domination was introduced by Cockayne et al."
"Various aspects of secure domination have been researched."
"A vertex v is a cut-vertex if G - {v} is disconnected."
"A set S is a dominating set if each vertex u ∈ V (G) \ S is adjacent to some vertex in S."
"A maximal outerplane graph without internal triangles is called stripped."