arXiv subjects

Miriam Münch

Publications and source records attributed to Miriam Münch.

4 recordsLinked to original sources

arXiv2026

Towards the Recognition of Oriented Interval Graphs

Oriented 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 $φ\colon V(G) \to \{-1,1\}$ of all intervals, (ii) a clique ordering $σ$, and (iii) a set $E_\mathrm{cont} \subseteq E(G)$ 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. In 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 $E_\mathrm{cont}$ 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.

cs.CG↗

arXiv2026

Grounded String Representations of Series-Parallel Graphs without Transitive Edges

In a {\em grounded string representation} of a graph there is a horizontal line $\ell$ and each vertex is represented as a simple curve below $\ell$ with one end point on $\ell$ such that two curves intersect if and only if the respective vertices are adjacent. A grounded string representation is a {\em grounded L-reverseL-representation} if each vertex is represented by a 1-bend orthogonal polyline. It is a {\em grounded L-representation} if in addition all curves are L-shaped. We show that every biconnected series-parallel graph without edges between the two vertices of a separation pair (i.e., {\em transitive edges}) admits a grounded L-reverseL-representation if and only if it admits a grounded string representation. Moreover, we can test in linear time whether such a representation exists. We also construct a biconnected series-parallel graph without transitive edges that admits a grounded L-reverseL-representation, but no grounded L-representation.

cs.CG↗

arXiv2025

Simple Realizability of Abstract Topological Graphs

An abstract topological graph (AT-graph) is a pair $A=(G,\mathcal{X})$, where $G=(V,E)$ is a graph and $\mathcal{X} \subseteq {E \choose 2}$ is a set of pairs of edges of $G$. A realization of $A$ is a drawing $Γ_A$ of $G$ in the plane such that any two edges $e_1,e_2$ of $G$ cross in $Γ_A$ if and only if $(e_1,e_2) \in \mathcal{X}$; $Γ_A$ is simple if any two edges intersect at most once (either at a common endpoint or at a proper crossing). The AT-graph Realizability (ATR) problem asks whether an input AT-graph admits a realization. The version of this problem that requires a simple realization is called Simple AT-graph Realizability (SATR). It is a classical result that both ATR and SATR are NP-complete. In this paper, we study the SATR problem from a new structural perspective. More precisely, we consider the size $\mathrmλ(A)$ of the largest connected component of the crossing graph of any realization of $A$, i.e., the graph ${\cal C}(A) = (E, \mathcal{X})$. This parameter represents a natural way to measure the level of interplay among edge crossings. First, we prove that SATR is NP-complete when $\mathrmλ(A) \geq 6$. On the positive side, we give an optimal linear-time algorithm that solves SATR when $\mathrmλ(A) \leq 3$ and returns a simple realization if one exists. Our algorithm is based on several ingredients, in particular the reduction to a new embedding problem subject to constraints that require certain pairs of edges to alternate (in the rotation system), and a sequence of transformations that exploit the interplay between alternation constraints and the SPQR-tree and PQ-tree data structures to eventually arrive at a simpler embedding problem that can be solved with standard techniques.

cs.DS↗

arXiv2022

Partial and Simultaneous Transitive Orientations via Modular Decomposition

A natural generalization of the recognition problem for a geometric graph class is the problem of extending a representation of a subgraph to a representation of the whole graph. A related problem is to find representations for multiple input graphs that coincide on subgraphs shared by the input graphs. A common restriction is the sunflower case where the shared graph is the same for each pair of input graphs. These problems translate to the setting of comparability graphs where the representations correspond to transitive orientations of their edges. We use modular decompositions to improve the runtime for the orientation extension problem and the sunflower orientation problem to linear time. We apply these results to improve the runtime for the partial representation problem and the sunflower case of the simultaneous representation problem for permutation graphs to linear time. We also give the first efficient algorithms for these problems on circular permutation graphs.

cs.DS↗