| 4 | Closed 3-stop Center and Periphery in Graphs显示文摘A delivery person must leave the central location of the business, deliver packages at a number of addresses, and then return. Naturally, he/she wishes to reduce costs by finding the most efficient route. This motivates the following: Given a set of k distinct vertices S = { x1, x2, . . . , xk } in a simple graph G, the closed k-stop-distance of set S is defined to be dk( S) = min θ∈P (S ) (d(θ(x1), θ(x2)) + d(θ(x2), θ(x3)) + ··· + d(θ(xk), θ(x1))), where P (S) is the set of all permutations of S. That is the same as saying that dk( S) is the length of a shortest closed walk through the vertices { x1, . . . , xk }. The closed 2-stop distance is twice the standard distance between two vertices. We study the closed k-stop center and closed k-stop periphery of a graph, for k = 3. | Linda EROH Ralucca GERA Steven J. WINTERS | 2012 | Acta Mathematica Sinica,English Series2012，28，3： | 0 |
| 5 | A Comparison between the Metric Dimension and Zero Forcing Number of Trees and Unicyclic Graphs显示文摘The metric dimension dim(G)of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices.The zero forcing number Z(G)of a graph G is the minimum cardinality of a set S of black vertices(whereas vertices in V(G)\S are colored white)such that V(G)is turned black after finitely many applications of'the color-change rule':a white vertex is converted black if it is the only white neighbor of a black vertex.We show that dim(T)≤Z(T)for a tree T,and that dim(G)≤Z(G)+1 if G is a unicyclic graph;along the way,we characterize trees T attaining dim(T)=Z(T).For a general graph G,we introduce the'cycle rank conjecture'.We conclude with a proof of dim(T)-2≤dim(T+e)≤dim(T)+1 for e∈E(T). | Linda EROH Cong X.KANG Eunjeong YI | 2017 | Acta Mathematica Sinica,English Series2017，33，6： | 0 |
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