Geodesic
The Regulation Number of a Graph
Publications de l'Institut Mathématique, _N_S_34 (1983) no. 48, p. 3

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The regulation number $r(G)$ of a graph $G$ with maximum degree $d$ is defined as the smallest number of new points in a $d$-regular supergraph. It is shown that for $d\geq3$, every possible value of $r(G)$ between zero and the maximum established by Akiyama, Era and Harary, namely, $d$(mod 2)$+1+d$, is realized by some graph. Also, a characterization is given for $G$ to have $r(G)=n$.
Classification : 03B50 
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     author = {Jin Akiyama and Frank Harary},
     title = {The {Regulation} {Number} of a {Graph}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {3 },
     publisher = {mathdoc},
     volume = {_N_S_34},
     number = {48},
     year = {1983},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1983_N_S_34_48_a0/}
}
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Jin Akiyama; Frank Harary. The Regulation Number of a Graph. Publications de l'Institut Mathématique, _N_S_34 (1983) no. 48, p. 3 . http://geodesic.mathdoc.fr/item/PIM_1983_N_S_34_48_a0/