For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W-tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steiner set containing T. A forcing subset for W of minimum cardinality is a minimum forcing subset of W. The...

Rainbow connection number of Cartesian products and their subgraphs are considered. Previously known bounds are compared and non-existence of such bounds for subgraphs of products are discussed. It is shown that the rainbow connection number of an isometric subgraph of a hypercube is bounded above by the rainbow connection number of the hypercube. Isometric subgraphs of hypercubes with the rainbow connection number as small as possible compared to the rainbow connection of the hypercube are constructed....

Let G be a connected graph. A vertex w ∈ V.G/ strongly resolves two vertices u,v ∈ V.G/ if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. It is well known that the problem of computing this invariant is NP-hard. In this paper we study...

Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give...

The paper describes the properties of two transformations of graphs. One of them was introduced by F. Gliviak for the sake of study of metric properties of graphs, the other is related to it.

For a connected graph $G$ of order $n\ge 2$ and a linear ordering $s:v_1,v_2,...,v_n$ of vertices of $G$, $d\left(s\right)={\sum }_{i=1}^{n-1}d(v_i,v_{i+1})$, where $d(v_i,v_{i+1})$ is the distance between $v_i$ and $v_{i+1}$. The upper traceable number $t^{+}\left(G\right)$ of $G$ is $t^{+}\left(G\right)=max\left\{d\left(s\right)\right\}$, where the maximum is taken over all linear orderings $s$ of vertices of $G$. It is known that if $T$ is a tree of order $n\ge 3$, then $2n-3\le t^{+}\left(T\right)\le \lfloor n^2/2\rfloor -1$ and $t^{+}\left(T\right)\le \lfloor n^2/2\rfloor -3$ if $T\ne P_n$. All pairs $n,k$ for which there exists a tree $T$ of order $n$ and $t^{+}\left(T\right)=k$ are determined and a characterization of all those trees of order $n\ge 4$ with upper traceable number $\lfloor n^2/2\rfloor -3$ is established. For a connected graph $G$ of order...

The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K₁ by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs and τ-graphs.

The reverse Wiener index of a connected graph $G$ is defined as $$\Lambda \left(G\right)=\frac{1}{2}n(n-1)d-W\left(G\right),$$ where $n$ is the number of vertices, $d$ is the diameter, and $W\left(G\right)$ is the Wiener index (the sum of distances between all unordered pairs of vertices) of $G$. We determine the $n$-vertex non-starlike trees with the first four largest reverse Wiener indices for $n\ge 8$, and the $n$-vertex non-starlike non-caterpillar trees with the first four largest reverse Wiener indices for $n\ge 10$.

The orientation distance graph 𝓓ₒ(G) of a graph G is defined as the graph whose vertex set is the pair-wise non-isomorphic orientations of G, and two orientations are adjacent iff the reversal of one edge in one orientation produces the other. Orientation distance graphs was introduced by Chartrand et al. in 2001. We provide new results about orientation distance graphs and simpler proofs to existing results, especially with regards to the bipartiteness of orientation distance graphs and the representation...