A graph and its complement with specified properties. I: Connectivity.
01 Jan 1979-International Journal of Mathematics and Mathematical Sciences (Hindawi Publishing Corporation, New York)-Vol. 2, Iss: 2, pp 223-228
TL;DR: In this paper, the conditions under which both a graph G and its complement G¯ possess a specified property were investigated, and all graphs G for which G and G& #175 have connectivity one, (b) have line-connectivity one, and (c) are 2-connected, (d) are forests, (e) are bipartite, (f) are outerplanar and (g) are eulerian.
Abstract: We investigate the conditions under which both a graph G and its complement G¯ possess a specified property. In particular, we characterize all graphs G for which G and G¯ both (a) have connectivity one, (b) have line-connectivity one, (c) are 2-connected, (d) are forests, (e) are bipartite, (f) are outerplanar and (g) are eulerian. The proofs are elementary but amusing.
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TL;DR: In 2005, Zhang and Wu studied the Nordhaus–Gaddum problem for the Wiener index and obtained analogous results for S W k, namely sharp upper and lower bounds for S w k + S W w k ( G ) and S w w ( G) ⋅ S Wk ( G ¯ ) , valid for any connected graph G whose complement G ¯ is also connected.
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...Akiyama and Harary [1] characterized the graphs for which both G and G are connected....
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TL;DR: The Nordhaus–Gaddum-type result for the conflict-free connection number of graphs is obtained, and it is proved that if G and G ¯ are connected graphs of order n ( n ≥ 4), then 4 ≤ c f c ( G ) + c fc ( G ¯ ) ≤ n and 4 ≥ 2 ( n − 2 ).
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Abstract: For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. In this paper, we summarize the known results on the Steiner distance parameters, including Steiner distance, Steiner diameter, Steiner center, Steiner median, Steiner interval, Steiner distance hereditary graph, Steiner distance stable graph, average Steiner distance, and Steiner Wiener index. It also contains some conjectures and open problems for further studies.
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TL;DR: This series characterize all the graphs G such that both G and G have the same number of endpoints, and finds that this number can only be 0 or 1 or 2, and is able to enumerate the self-complementary blocks.
Abstract: In this series, we investigate the conditions under which both a graph G and its complement G possess certain specified properties. We now characterize all the graphs G such that both G and G have the same number of endpoints, and find that this number can only be 0 or 1 or 2. As a consequence, we are able to enumerate the self-complementary blocks.
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