Distance domination, guarding and covering of maximal outerplanar graphs

Abstract

a b s t r a c tIn this paper we introduce the notion of distance k-guarding applied to triangulation graphs, and associate it with distance k-domination and distance k-covering. We obtain results for maximal outerplanar graphs when k = 2. A set S of vertices in a triangulation graph T is a distance 2-guarding set (or 2d-guarding set for short) if every face of T has a vertex adjacent to a vertex of S. We show that ⌊ n 5 ⌋ (respectively, ⌊ n 4 ⌋) vertices are sufficient to 2d-guard and 2d-dominate (respectively, 2d-cover) any n-vertex maximal outerplanar graph. We also show that these bounds are tight.

Monitoring maximal outerplanar graphs

“…First, we start by presenting the terminology that we use when the elements of T are monitored by vertices, at its distance version (see [2], for details). Let T = (V, E) be a triangulation.…”

“…The presence of any of the internal edges (0,7), (0,6), (0,5), (7,1), (7,2) and (7,3) would violate the minimality of m. Thus, the triangle T ′ in T 1 that is bounded by e is (0,4,8). Consider the maximal outerplanar graph T * = T 2 + (0, 4, 5, 6, 7, 8) (see Fig.3(a)).…”

“…An edge guarding set L ⊆ E is a set which guards every face in G. The edge guarding number, g e (G), is the minimum cardinality of an edge guarding set for G. All the previous described monitoring notions were extended to include its distance versions on plane graphs. For example, domination was extended to distance domination and guarding to distance guarding [2].…”

<i>γ</i><i>k</i>(<i>n</i>) = max {⌊<i>n</i>/(2<i>k</i>+1)⌋, 1} for Maximal Outerplanar Graphs with <i>n</i> mod (2<i>k</i>+1) ≤ 6

“…Instead there is a conjecture γ k (n) = n 2k+1 , proved for k = 1, 2 (Refs. [1], [10]). In this paper, we give a unified and simpler proof for k = 1, 2, 3.…”

Combinatorial bounds on connectivity for dominating sets in maximal outerplanar graphs

“…In recent years it has received special attention the problem of domination in outerplanar graphs (e.g., [1,2,11]) A graph is outerplanar if it has a crossingfree embedding in the plane such th a t all vertices are on the boundary of its outer face (the unbounded face). An outerplanar graph is maximal if it is not possible to add an edge such th a t the resulting graph is still outerplanar.…”