Extending Partial Representations of Circle Graphs in Near-Linear Time

The partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph H ⊆ G and a representation H of H. The question is whether G admits a representation G whose restriction to H is H. We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams. We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in O((n+m)α(n+m)) time, where α is the inverse Ackermann function. This improves over an O(n$^{3}$)-time algorithm by Chaplick, Fulek and Klavík [2019]. The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest

Bounded Representations of Interval and Proper Interval Graphs

Klavik et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals L_v and R_v called bounds. We ask whether there exists a bounded representation in which each interval I_v has its left endpoint in L_v and its right endpoint in R_v. We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs. Robert's Theorem states that the classes of proper interval graphs and unit interval graphs are equal. Surprisingly the bounded representation problem is polynomially solvable for proper interval graphs and NP-complete for unit interval graphs [Klav\'{\i}k et al., arxiv:1207.6960]. So unless P = NP, the proper and unit interval representations behave very differently. The bounded representation problem belongs to a wider class of restricted representation problems. These problems are generalizations of the well-understood recognition problem, and they ask whether there exists a representation of G satisfying some additional constraints. The bounded representation problems generalize many of these problems

Minimal Obstructions for Partial Representations of Interval Graphs

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension

Dotykové grafy kružnic a Möbiovy transformace

A graph can be represented by various geometric representations. In this work we focus on the circle packing representation. We state various concepts impor- tant for proving results regarding this kind of representation. We introduce a known proof of existence of a circle packing for planar graphs and a proof of existence of a primal-dual circle packing for 3-connected graphs. Next, we focus on computational complexity of extending the representation for a given partial circle packing. We examine the proof of the theorem stating that deciding whet- her such an extension exists is an NP-hard problem. We introduce our theoretical algorithm for extension construction based on real RAM machine. 1Graf lze reprezentovat různými geometrickými reprezentacemi. V této práci se věnujeme reprezentaci grafů pomocí circle packingu (dotykových kružnic). Roze- bereme důležité koncepty potřebné pro dokázání klíčových výsledků ohledně této reprezentace. Představíme konkrétní známý důkaz existence circle packingu pro rovinné grafy a existence primal-dual circle packingu pro 3-souvislé grafy. Dále se budeme zabývat složitostí problému rozšíření reprezentace při zadaném čás- tečném circle packingu. Rozebereme důkaz věty, která říká, že rozhodnout, zda lze nalézt takové rozšíření, je NP-těžký problém. Představíme vlastní teoretický algoritmus pro konstrukci rozšíření založený na real RAM stroji. 1Katedra aplikované matematikyDepartment of Applied MathematicsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic