Lucille Bugo - Academia.edu (original) (raw)

Papers by Lucille Bugo

Research paper thumbnail of On sequential geodetic numbers of some connected graphs

Applied Mathematical Sciences, 2014

Given two vertices u and v of a connected graph G, the closed interval I G [u, v] is that set of ... more Given two vertices u and v of a connected graph G, the closed interval I G [u, v] is that set of all vertices lying in some u-v geodesic in G. If S ⊆ V (G), then I G [S] = ∪{I G [u, v] : u, v ∈ S}. Let v i ∈ V (G) for i = 1, 2, ..., n. We select vertices of G as follows: Select v 1 and let S 1 = {v 1 }. Select another vertex v 2 = v 1 and let S 2 = {v 1 , v 2 }. Then successively select vertex v k / ∈ S k−1 and let S k = S k−1 ∪ {v k } ∪ {u ∈ V (G) : u ∈ I G [v k , w] for some w ∈ S k−1 }. The sequential geodetic number of G, denoted by sgn(G) is the smallest k such that there is a sequence v 1 , v 2 , ..., v k for which S k = V (G). The set S = S k = {v 1 , v 2 ,. .. , v k } with v 1 , v 2 ,. .. , v k ∈ S k for which S k = V (G) is a sequential geodetic cover of G. The sequential geodetic number is again inspired by the achievement and avoidance games. In this paper, some connected graphs G with sgn(G) equals |V (G)|−1 and those equal to |V (G)| are characterized. It is shown that the geodetic number (gn), closed geodetic number (cgn) and the sequential geodetic number (sgn) coincide for some particular graphs. Further, for the complete bipartite graph K m,n these three graph invariants are determined.

Research paper thumbnail of On sequential geodetic numbers of graphs under some binary operations

Applied Mathematical Sciences, 2014

Given two vertices u and v of a connected graph G, the closed interval I G [u, v] is that set of ... more Given two vertices u and v of a connected graph G, the closed interval I G [u, v] is that set of all vertices lying in some u-v geodesic in G. If S ⊆ V (G), then I G [S] = ∪{I G [u, v] : u, v ∈ S}. Let v i ∈ V (G) for i = 1, 2, ..., n. We select vertices of G as follows: Select v 1 and let S 1 = {v 1 }. Select another vertex v 2 = v 1 and let S 2 = {v 1 , v 2 }. Then successively select vertex v k / ∈ S k−1 and let S k = S k−1 ∪ {v k } ∪ {u ∈ V (G) : u ∈ I G [v k , w] for some w ∈ S k−1 }. The sequential geodetic number of G, denoted by sgn(G) is the smallest k such that there is a sequence v 1 , v 2 , ..., v k for which S k = V (G). The set S = S k = {v 1 , v 2 ,. .. , v k } with v 1 , v 2 ,. .. , v k ∈ S k for which S k = V (G) is a sequential geodetic cover of G. The sequential geodetic number is again inspired by the achievement and avoidance games [11]. In this paper, the sequential geodetic numbers of graphs obtained from the join and corona of graphs are determined. This paper is the second part of [3]. 2 Sequential Geodetic Numbers of the Join of Graphs For the purpose of this study, the following definitions of a geodetic sequence, sequential geodetic cover and sequential geodetic basis are introduced. Definition 2.1 Let G be a connected graph. A sequence v 1 , v 2 ,. .. , v k of the vertices in G is a geodetic sequence if it generates a sequence S 1 , S 2 ,. .. , S k of subsets of V (G) satisfying the following: (1) v 1 = v 2 for which S 1 = {v 1 } and S 2 = {v 1 , v 2 }; (2) v i / ∈ S i−1 for 3 ≤ i ≤ k, with S i−1 = {v 1 , v 2 ,. .. , v i−1 } that determines v 1 , v 2 ,. .. , v k for which S i = S i−1 ∪ {v i } ∪ {u ∈ V (G) : u ∈ I G [v i , w] for some w ∈ S i−1 }.

Research paper thumbnail of On sequential geodetic numbers of some connected graphs

Applied Mathematical Sciences, 2014

Given two vertices u and v of a connected graph G, the closed interval I G [u, v] is that set of ... more Given two vertices u and v of a connected graph G, the closed interval I G [u, v] is that set of all vertices lying in some u-v geodesic in G. If S ⊆ V (G), then I G [S] = ∪{I G [u, v] : u, v ∈ S}. Let v i ∈ V (G) for i = 1, 2, ..., n. We select vertices of G as follows: Select v 1 and let S 1 = {v 1 }. Select another vertex v 2 = v 1 and let S 2 = {v 1 , v 2 }. Then successively select vertex v k / ∈ S k−1 and let S k = S k−1 ∪ {v k } ∪ {u ∈ V (G) : u ∈ I G [v k , w] for some w ∈ S k−1 }. The sequential geodetic number of G, denoted by sgn(G) is the smallest k such that there is a sequence v 1 , v 2 , ..., v k for which S k = V (G). The set S = S k = {v 1 , v 2 ,. .. , v k } with v 1 , v 2 ,. .. , v k ∈ S k for which S k = V (G) is a sequential geodetic cover of G. The sequential geodetic number is again inspired by the achievement and avoidance games. In this paper, some connected graphs G with sgn(G) equals |V (G)|−1 and those equal to |V (G)| are characterized. It is shown that the geodetic number (gn), closed geodetic number (cgn) and the sequential geodetic number (sgn) coincide for some particular graphs. Further, for the complete bipartite graph K m,n these three graph invariants are determined.

Research paper thumbnail of On sequential geodetic numbers of graphs under some binary operations

Applied Mathematical Sciences, 2014

Given two vertices u and v of a connected graph G, the closed interval I G [u, v] is that set of ... more Given two vertices u and v of a connected graph G, the closed interval I G [u, v] is that set of all vertices lying in some u-v geodesic in G. If S ⊆ V (G), then I G [S] = ∪{I G [u, v] : u, v ∈ S}. Let v i ∈ V (G) for i = 1, 2, ..., n. We select vertices of G as follows: Select v 1 and let S 1 = {v 1 }. Select another vertex v 2 = v 1 and let S 2 = {v 1 , v 2 }. Then successively select vertex v k / ∈ S k−1 and let S k = S k−1 ∪ {v k } ∪ {u ∈ V (G) : u ∈ I G [v k , w] for some w ∈ S k−1 }. The sequential geodetic number of G, denoted by sgn(G) is the smallest k such that there is a sequence v 1 , v 2 , ..., v k for which S k = V (G). The set S = S k = {v 1 , v 2 ,. .. , v k } with v 1 , v 2 ,. .. , v k ∈ S k for which S k = V (G) is a sequential geodetic cover of G. The sequential geodetic number is again inspired by the achievement and avoidance games [11]. In this paper, the sequential geodetic numbers of graphs obtained from the join and corona of graphs are determined. This paper is the second part of [3]. 2 Sequential Geodetic Numbers of the Join of Graphs For the purpose of this study, the following definitions of a geodetic sequence, sequential geodetic cover and sequential geodetic basis are introduced. Definition 2.1 Let G be a connected graph. A sequence v 1 , v 2 ,. .. , v k of the vertices in G is a geodetic sequence if it generates a sequence S 1 , S 2 ,. .. , S k of subsets of V (G) satisfying the following: (1) v 1 = v 2 for which S 1 = {v 1 } and S 2 = {v 1 , v 2 }; (2) v i / ∈ S i−1 for 3 ≤ i ≤ k, with S i−1 = {v 1 , v 2 ,. .. , v i−1 } that determines v 1 , v 2 ,. .. , v k for which S i = S i−1 ∪ {v i } ∪ {u ∈ V (G) : u ∈ I G [v i , w] for some w ∈ S i−1 }.