{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:00Z","timestamp":1753893780510,"version":"3.41.2"},"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>A unit bar-visibility graph is a graph whose vertices can be represented in the plane by disjoint horizontal unit-length bars such that two vertices are adjacent if and only if there is a unobstructed, non-degenerate, vertical band of visibility between the corresponding bars. We generalize unit bar-visibility graphs to $[1,k]$-bar-visibility graphs by allowing the lengths of the bars to be between $1\/k$ and $1$. We completely characterize these graphs for trees. We establish an algorithm with complexity $O(kn)$ to determine whether a tree with $n$ vertices has a $[1,k]$-bar-visibility representation. In the course of developing the algorithm, we study a special case of the knapsack problem: Partitioning a set of positive integers into two sets with sums as equal as possible. We give a necessary and sufficient condition for the existence of such a partition.<\/jats:p>","DOI":"10.37236\/1116","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:56:22Z","timestamp":1578700582000},"source":"Crossref","is-referenced-by-count":3,"title":["Characterization of $[1,k]$-Bar Visibility Trees"],"prefix":"10.37236","volume":"13","author":[{"given":"Guantao","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joan P.","family":"Hutchinson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ken","family":"Keating","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Shen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2006,10,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v13i1r90\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v13i1r90\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:07:20Z","timestamp":1579302440000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v13i1r90"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,10,27]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2006,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1116","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2006,10,27]]},"article-number":"R90"}}