[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81925-en":3,"doc-seo-81925-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},81925,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Towards the Recognition of Oriented Interval Graphs","Oriented interval graphs extend classical interval graphs by assigning each interval an orientation (left or right). Overlaps with the same orientation yield directed arcs, while nested intervals and overlaps with opposite orientations yield undirected edges, producing a mixed intersection graph. An oriented interval representation is combinatorially described by an orientation mapping φ, a clique ordering σ, and a set of containment edges Econt. The dependencies among these elements make recognition difficult; this work analyzes consistency conditions and provides linear-time algorithms for constrained recognition, improving earlier quadratic bounds and resolving the case of oriented proper/unit interval graphs.","arXiv :2607 .05 19 1v 1 [ cs .CG] 6 Jul 2026  \nTowards the Recognition of Oriented Interval Graphs  \nLukas P. Bachmann \\#  Universität Passau, Passau, Germany Jiří Fiala \\#   \nCharles University, Prague, Czech Republic Miriam Münch \\#   \nUniversität Passau, Passau, Germany Ignaz Rutter \\#  Universität Passau, Passau, Germany Peter Stumpf \\#   \nCharles University, Prague, Czech Republic Alexander Wolff Ñ   \nUniversität Würzburg, Würzburg, Germany  \n~~ Abstract ~~  \nOriented interval graphs, a recent generalization of interval graphs introduced by Gutowski et al. [GD 2022], are intersection graphs of intervals, each of which is oriented either left or right. Such a representation defines a mixed intersection graph: overlapping intervals with the same orientation define a (directed) arc; nested intervals (irrespective of the orientations of the intervals) and overlapping intervals of opposite orientations define an (undirected) edge. An oriented interval representation of a mixed graph G can be described combinatorially by the combination of (i) an orientation φ : V (G) → {−1, 1} of all intervals, (ii) a clique ordering σ, and (iii) a set Econt ⊆ E (G) of containment edges, which are represented by nested intervals. The non-trivial dependencies between these three ingredients make the recognition of oriented interval graphs a challenging problem.  \nIn this paper, we take steps towards a general recognition algorithm by studying how orientation, clique ordering, and containment edges influence and restrict each other. We characterize the orientations that are consistent with a given set of containment edges as well as the clique orderings that are consistent with a given orientation. Based on these characterizations, we give linear-time algorithms for two constrained versions of the recognition problem where, in addition to the mixed input graph G, either the set of containment edges Econt or the orientation φ is prescribed. This improves a quadratic-time algorithm of Gutowski et al. for the case that all vertices have the same orientation; an assumption that determines both the orientation and the containment edges. In particular, this also solves the recognition problem for oriented proper (or unit) interval graphs.  \n2012 ACM Subject Classification Mathematics of computing → Graph theory; Theory of compu  \ntation → Computational geometry  \nKeywords and phrases Interval graphs, mixed graphs, oriented interval graphs, recognition Funding Jiří Fiala: Supported by grant no. 25-16847S of the Czech Science Foundation (GAČR) .  \n 1  Introduction  \nA mixed graph is a graph that may contain both (undirected) edges and (directed) arcs. Mixed graphs were introduced by Sotskov and Tanaev [21] and reintroduced by Hansen et al. [14] and have been studied in the context of scheduling problems [20], (quasi-)upward planar drawings [2, 3, 8] and extensions of partial orientations [1] .  \nAn intersection representation of a graph G is a mapping ρ : V (G) → C from the vertex set V (G) of G, to a class C of geometric objects such that {u, v} is an edge of G if and only  \n2 Towards the Recognition of Oriented Interval Graphs  \nif ρ(u) ∩ ρ(v)  ∅ . Every class C of geometric objects gives rise to a class of graphs that admit an intersection representation with objects in C. Prominent examples include interval graphs (intersection graphs of intervals on the real line), circular-arc graphs (intersection graphs of arcs on a circle), circle graphs (intersection graphs of chords of a circle), and permutation graphs (intersection graphs of line segments connecting two parallel lines) .  \nInterval graphs form a well-understood class in structural and algorithmic graph theory, with applications spanning from scheduling problems to genome analysis [12] . Using so-called PQ-trees, one can test in linear time whether a graph G is an interval graph and, if yes, compute an interval representation of G [4] . Some problems, such as coloring or maximum clique, that a","cbCaimMfkS8Qi7RD","https://ap.wps.com/l/cbCaimMfkS8Qi7RD","pdf",1777764,1,31,"English","en",105,"# Introduction\n## Oriented interval graphs and their mixed intersection structure\n## Combinatorial description via φ, σ, and containment edges\n## Recognition challenges and prior work\n## Contributions and linear-time constrained algorithms","[{\"question\":\"How are oriented interval graphs defined from interval representations?\",\"answer\":\"Each interval is oriented left or right. Overlapping intervals with the same orientation create directed arcs, while nested intervals (regardless of orientation) and overlaps of opposite orientations create undirected edges.\"},{\"question\":\"What combinatorial components describe an oriented interval representation?\",\"answer\":\"A representation is described by (i) an orientation mapping φ from vertices to {−1, 1}, (ii) a clique ordering σ, and (iii) a set of containment edges Econt represented by nested intervals.\"},{\"question\":\"What does the paper contribute toward recognition algorithms?\",\"answer\":\"It studies how orientation, clique ordering, and containment edges constrain each other, characterizes consistent choices, and derives linear-time algorithms for recognition variants where either Econt or φ is prescribed.\"}]",1784177081,78,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"towards-the-recognition-of-oriented-interval-graphs","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/towards-the-recognition-of-oriented-interval-graphs/81925/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"How are oriented interval graphs defined from interval representations?","Question",{"text":74,"@type":75},"Each interval is oriented left or right. Overlapping intervals with the same orientation create directed arcs, while nested intervals (regardless of orientation) and overlaps of opposite orientations create undirected edges.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What combinatorial components describe an oriented interval representation?",{"text":79,"@type":75},"A representation is described by (i) an orientation mapping φ from vertices to {−1, 1}, (ii) a clique ordering σ, and (iii) a set of containment edges Econt represented by nested intervals.",{"name":81,"@type":72,"acceptedAnswer":82},"What does the paper contribute toward recognition algorithms?",{"text":83,"@type":75},"It studies how orientation, clique ordering, and containment edges constrain each other, characterizes consistent choices, and derives linear-time algorithms for recognition variants where either Econt or φ is prescribed.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]