{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T01:53:36Z","timestamp":1767837216234,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>An unweighted, connected graph is distance-balanced (also called self-median) if there exists a number $d$ such that, for any vertex $v$, the sum of the distances from $v$ to all other vertices is $d$.  An unweighted connected graph is strongly distance-balanced (also called distance-degree regular) if there exist numbers $d_1,d_2,d_3,\\dots$ such that, for any vertex $v$, there are precisely $d_k$ vertices at distance $k$ from $v$. We consider the following optimization problem: given a graph, add the minimum possible number of edges to obtain a (strongly) distance-balanced graph.  We show that the problem is NP-hard for graphs of diameter three, thus answering the question posed by Jerebic et al. [Distance-balanced graphs; Ann. Comb. 2008].  In contrast, we show that the problem can be solved in polynomial time for graphs of diameter 2.<\/jats:p>","DOI":"10.37236\/536","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:52:59Z","timestamp":1578714779000},"source":"Crossref","is-referenced-by-count":14,"title":["The Complexity of Obtaining a Distance-Balanced Graph"],"prefix":"10.37236","volume":"18","author":[{"given":"Sergio","family":"Cabello","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Primo\u017e","family":"Luk\u0161i\u010d","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,2,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p49\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:15:58Z","timestamp":1579302958000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/536","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,2,28]]},"article-number":"P49"}}