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### On strong distances in oriented graphs

Peter Dankelmann, Henda C. Swart, David P. Day
2003 Discrete Mathematics
For asymmetric digraphs (that is, oriented graphs) we present bounds on the strong radius in terms of order and on the strong diameter in terms of order, girth and connectivity.  ...  The strong distance between two vertices u and v in D, denoted by sdD(u; v) is the minimum size of a strongly connected subdigraph of D containing u and v.  ...  Bounds on strong diameter In this section we present bounds on the strong diameter of a strong oriented graph D.  ...

### Page 7614 of Mathematical Reviews Vol. , Issue 2000k [page]

2000 Mathematical Reviews
strong distance in strong oriented graphs.  ...  It is shown that every oriented graph is the strong center of some strong oriented graph. A strong oriented graph D is called strongly self-centered if D is its own strong center.  ...

### Lower and upper orientable strong diameters of graphs satisfying the Ore condition

Meirun Chen, Xiaofeng Guo, Hao Li
2009 Applied Mathematics Letters
In this work, we determine a bound of the lower orientable strong diameters and the bounds of the upper orientable strong diameters for graphs G = (V , E) satisfying the Ore condition (that is, σ 2 (G)  ...  Lower and upper orientable strong diameter Lower and upper orientable strong radius Strong distance Strong eccentricity Strong radius and strong diameter The Ore condition a b s t r a c t Let D be a strong  ...  Let D be a strong oriented graph and κ(D) = κ. Then sdiam(D) In , Dankelmann et al. also gave an upper bound on the strong radius of a strong oriented graph D. Theorem 6 ().  ...

### On \$k\$-strong distance in strong digraphs

Ping Zhang
2002 Mathematica Bohemica
It was shown in  that strong distance is a metric on the vertex set of a strong oriented graph D. As such, certain properties are satisfied.  ...  In the strong oriented graph D of Figure 1 , sd(v, w) = 3, sd(u, y) = 4, and sd(u, x) = 5. A generalization of distance in graphs was introduced in  .  ...

### Page 5313 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews
In addition, some new char- acterizations as meshed graphs and one in terms of location theory are presented.  ...  It is shown that every pair r, d of integers with 3 <r <d <2r is realizable as the strong radius and strong diameter of some strong oriented graph.  ...

### Minimum average distance of strong orientations of graphs

P DANKELMANN
2004 Discrete Applied Mathematics
If G is a 2-edge-connected graph, then˜ min (G) is the minimum average distance taken over all strong orientations of G.  ...  The average distance of a graph (strong digraph) G, denoted by (G) is the average, among the distances between all pairs (ordered pairs) of vertices of G.  ...  For a 2-edge-connected graph G,˜ min (G) is the minimum of the average distances of strong orientations of G taken over all strong orientations of G.  ...

### Minimum average distance of strong orientations of graphs

Peter Dankelmann, Ortrud R. Oellermann, Jian-Liang Wu
2004 Discrete Applied Mathematics
If G is a 2-edge-connected graph, then˜ min (G) is the minimum average distance taken over all strong orientations of G.  ...  The average distance of a graph (strong digraph) G, denoted by (G) is the average, among the distances between all pairs (ordered pairs) of vertices of G.  ...  For a 2-edge-connected graph G,˜ min (G) is the minimum of the average distances of strong orientations of G taken over all strong orientations of G.  ...

### The optimal strong radius and optimal strong diameter of the Cartesian product graphs

Meirun Chen, Xiaofeng Guo, Shaohui Zhai
2011 Applied Mathematics Letters
[Justie Su-Tzu Juan, Chun-Ming Huang, I-Fan Sun, The strong distance problem on the Cartesian product of graphs, Inform. Process.  ...  The optimal strong radius (resp. strong diameter) srad(G) (resp. sdiam(G)) of a graph G is the minimum strong radius (resp. strong diameter) over all strong orientations of G.  ...  Some known results on the strong distance, strong radius and strong diameter can be found in    .  ...

### Page 5745 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews
In this paper it is shown that, for each integer k > 2, every oriented graph is the k-strong center of some strong digraph.  ...  Martin Knor (SK-STU; Bratislava) 2003h:05075 05Ci2 05C20 Zhang, Ping |Zhang, Ping®] (1-WMI-DM; Kalamazoo, MI) On k-strong distance in strong digraphs. (English summary) Math.  ...

### Page 4745 of Mathematical Reviews Vol. , Issue 92i [page]

1992 Mathematical Reviews
It is proven in this paper that for any two oriented graphs D, and D, and positive integer k, there exists a strong oriented graph H whose m-center and m-median are isomorphic to D, and D>, respectively  ...  i(C) is an orientation-preserving curve on S.  ...

### Note on the relation between radius and diameter of a graph

Ferdinand Gliviak, Peter Kyš
1995 Mathematica Bohemica
Paper  notes that if we have any distance on graphs which is a metric and then define eccentricity, radius, and diameter as usual in terms of this metric, then the corresponding inequality holds.  ...  In this note we prove that for any natural numbers r ^ 1, d > 2r there exists a strong digraph D with radius r and diameter d. (Distance and eccentricity are the usual ones.)  ...

### Strong embeddings of minimum genus

Bojan Mohar
2010 Discrete Mathematics
Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every bridgeless cubic graph can be embedded on a surface of its own genus in such a way that the face boundaries are cycles of the graph  ...  to the order of these graphs), thus providing plethora of strong counterexamples to the above conjecture.  ...  This is no longuer true on the torus. Figure 1 shows two embeddings of a non-planar cubic graph in the torus. One is strong, and the other one is not.  ...

### Shattering, Graph Orientations, and Connectivity [article]

Laszlo Kozma, Shay Moran
2012 arXiv   pre-print
These examples are derived from properties of orientations related to distances and flows in networks.  ...  In one direction, we use this connection to derive results in graph theory. Our main tool is a generalization of the Sauer-Shelah Lemma.  ...  The work developed from results in the Master thesis of Shay Moran; we would like to acknowledge again the contribution of the advisor of this thesis -Ami Litman.  ...

### Oriented graphs with prescribed \$m\$-center and \$m\$-median

Gary Chartrand, Song Lin Tian
1991 Czechoslovak Mathematical Journal
Oriented graphs with prescribed m-center and m-median  ...  The maximum distance or m-distance md{u, v) between u and v is max {d(u,v), d(v, u)}. It is not difficult to show that the m-distance is a metric on the vertex set of a strong digraph.  ...  For vertices u and v in a strong digraph D, the directed distance d(u, v) is the length ofa shortest (directed) u -v path in D.  ...

### Page 5774 of Mathematical Reviews Vol. , Issue 95j [page]

1995 Mathematical Reviews
(SA-NTL; Durban) On the integrity of distance domination in graphs. (English summary) Australas. J. Combin. 10 (1994), 29-43.  ...  In this paper, we show that the strong embedding conjecture for chain graphs is equivalent to the strong embed- ding conjecture.  ...
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