We present an algorithm that determines the prime factors of connected graphs with respect to the Cartesian product in linear time and space. This improves a result of Aurenhammer et al. [Cartesian graph factorization at logarithmic cost per edge, Comput. Complexity 2 (1992) 331-349], who compute the prime factors in O(m log n) time, where m denotes the number of vertices of G and n the number of edges. Our algorithm is conceptually simpler. It gains its efficiency by the introduction of edge-labellings.
A graph G is strongly distance-balanced if for every edge uv of G and every i ≥ 0 the number of vertices x with d(x, u) = d(x, v) − 1 = i equals the number of vertices y with d(y, v) = d(y, u) − 1 = i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distancebalanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O(mn) for their recognition, where m is the number of edges and n the number of vertices of the graph in question, are given.
Let G = (V, E) be a connected graph. Given a vertex v ∈ V and an edge e = uw ∈ E, the distance between v and e is defined as d G (e, v) = min{d G (u, v), d G (w, v)}. A nonempty set S ⊂ V is an edge metric generator for G if for any two edges e 1 , e 2 ∈ E there is a vertex w ∈ S such that d G (w, e 1 ) = d G (w, e 2 ). The minimum cardinality of any edge metric generator for a graph G is the edge metric dimension of G. The edge metric dimension of the join, lexicographic and corona product of graphs is studied in this article.
a b s t r a c tA walk W between two non-adjacent vertices in a graph G is called tolled if the first vertex of W is among vertices from W adjacent only to the second vertex of W , and the last vertex of W is among vertices from W adjacent only to the second-last vertex of W . In the resulting interval convexity, a set S ⊂ V (G) is toll convex if for any two non-adjacent vertices x, y ∈ S any vertex in a tolled walk between x and y is also in S. The main result of the paper is that a graph is a convex geometry (i.e. satisfies the Minkowski-Krein-Milman property stating that any convex subset is the convex hull of its extreme vertices) with respect to toll convexity if and only if it is an interval graph. Furthermore, some well-known types of invariants are studied with respect to toll convexity, and toll convex sets in three standard graph products are completely described.
Let G = (V, E) be a graph of order n. A distance magic labeling of G is a bijection : V → {1, . . . , n} for which there exists a positive integer k such thatWe introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second factor for lexicographic product with regular graphs. In addition, we characterize distance magic graphs among direct product of two cycles.
Given a connected graph G, a vertex w ∈ V (G) distinguishes two different vertices u, v of G if the distances between w and u and between w and v are different. Moreover, w strongly resolves the pair u, v if there exists some shortest u − w path containing v or some shortest v − w path containing u. A set W of vertices is a (strong) metric generator for G if every pair of vertices of G is (strongly resolved) distinguished by some vertex of W . The smallest cardinality of a (strong) metric generator for G is called the (strong) metric dimension of G. In this article we study the (strong) metric dimension of some families of direct product graphs.
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