Computing directed Steiner path covers

In this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph $$G=(V,E)$$ G = ( V , E ) and a set $$T \subseteq V$$ T ⊆ V of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

Strictifying and taming directed paths in Higher Dimensional Automata

Directed paths have been used by several authors to describe concurrent executions of a program. Spaces of directed paths in an appropriate state space contain executions with all possible legal schedulings. It is interesting to investigate whether one obtains different topological properties of such a space of executions if one restricts attention to schedulings with “nice” properties, e.g. involving synchronisations. This note shows that this is not the case, i.e. that one may operate with nice schedulings without inflicting any harm. Several of the results in this note had previously been obtained by Ziemiański in Ziemiański (2017. Applicable Algebra in Engineering, Communication and Computing28 497–525; 2020a. Journal of Applied and Computational Topology4 (1) 45–78). We attempt to make them accessible for a wider audience by giving an easier proof for these findings by an application of quite elementary results from algebraic topology; notably the nerve lemma.

Subpath Queries on Compressed Graphs: A Survey

Text indexing is a classical algorithmic problem that has been studied for over four decades: given a text T, pre-process it off-line so that, later, we can quickly count and locate the occurrences of any string (the query pattern) in T in time proportional to the query’s length. The earliest optimal-time solution to the problem, the suffix tree, dates back to 1973 and requires up to two orders of magnitude more space than the plain text just to be stored. In the year 2000, two breakthrough works showed that efficient queries can be achieved without this space overhead: a fast index be stored in a space proportional to the text’s entropy. These contributions had an enormous impact in bioinformatics: today, virtually any DNA aligner employs compressed indexes. Recent trends considered more powerful compression schemes (dictionary compressors) and generalizations of the problem to labeled graphs: after all, texts can be viewed as labeled directed paths. In turn, since finite state automata can be considered as a particular case of labeled graphs, these findings created a bridge between the fields of compressed indexing and regular language theory, ultimately allowing to index regular languages and promising to shed new light on problems, such as regular expression matching. This survey is a gentle introduction to the main landmarks of the fascinating journey that took us from suffix trees to today’s compressed indexes for labeled graphs and regular languages.

Generalized Spectral Characterization of Mixed Graphs

The spectral characterization of graphs is an important topic in spectral graph theory, which has been studied extensively in recent years. Unlike the undirected case, however, the spectral characterization of mixed graphs (digraphs) has received much less attention so far, which will be the main focus of this paper. A mixed graph $G$ is said to be strongly determined by its generalized Hermitian spectrum (abbreviated SHDGS), if, up to isomorphism, $G$ is the unique mixed graph that is cospectral with $G$ w.r.t. the generalized Hermitian spectrum. Let $G$ be a self-converse mixed graph of order $n$ with Hermitian adjacency matrix $A$ and let $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector). Suppose that $2^{-\lfloor n/2\rfloor}\det W$ is \emph{norm-free} in $\mathbb{Z}[i]$ (i.e., for any Gaussian prime $p$, the norm $N(p)=p\bar{p}$ does not divide $2^{-\lfloor n/2\rfloor}\det W$). We conjecture that every such graph is SHDGS and prove that, for any mixed graph $H$ that is cospectral with $G$ w.r.t. the generalized Hermitian spectrum, there exists a Gaussian rational unitary matrix $U$ with $Ue=e$ such that $U^*A(G)U=A(H)$ and $(1+i)U$ is a Gaussian integral matrix. We have verified the conjecture in two extremal cases when $G$ is either an undirected graph or a self-converse oriented graph. Moreover, as consequences of our main results, we prove that all directed paths of even order are SHDGS. Analogous results are also obtained in the setting of \emph{restrictive} determination by generalized Hermitian spectrum (i.e., the spectral determination within the subset of all self-converse mixed graphs), which extends a recent result of the first author on the generalized spectral characterization of undirected graphs.

Inessential directed maps and directed homotopy equivalences

A directed space is a topological space $X$ together with a subspace $\vec {P}(X)\subset X^I$ of directed paths on $X$ . A symmetry of a directed space should therefore respect both the topology of the underlying space and the topology of the associated spaces $\vec {P}(X)_-^+$ of directed paths between a source ( $-$ ) and a target ( $+$ )—up to homotopy. If it is, moreover, homotopic to the identity map—in a directed sense—such a symmetry will be called an inessential d-map, and the paper explores the algebra and topology of inessential d-maps. Comparing two d-spaces $X$ and $Y$ ‘up to symmetry’ yields the notion of a directed homotopy equivalence between them. Under appropriate conditions, all directed homotopy equivalences are shown to satisfy a 2-out-of-3 property. Our notion of directed homotopy equivalence does not agree completely with the one defined in Goubault (2017, arxiv:1709:05702v2) and Goubault, Farber and Sagnier (2020, J. Appl. Comput. Topol. 4, 11–27); the deviation is motivated by examples. Nevertheless, directed topological complexity, introduced in Goubault, Farber and Sagnier (2020) is shown to be invariant under our notion of directed homotopy equivalence. Finally, we show that directed homotopy equivalences result in isomorphisms on the pair component categories of directed spaces introduced in Goubault, Farber and Sagnier (2020).

Linial's Dual Conjecture for Path-Spine Digraphs

Given a digraph D, a coloring 𝒞 of D is a partition of V(D) into stable sets. The k-norm of 𝒞 is defined as ΣC∈𝒞 min{|C|, k}. A coloring of D with minimum k-norm has its k-norm noted by χk(D). A (path)-k-pack of a digraph D is a set of k vertex-disjoint (directed) paths of D. The weight of a k-pack is the number of vertices covered by the k-pack. We denote by λk(D) the weight of a maximum k-pack. Linial conjectured that χk(D) ≤ λk(D) for every digraph. Such conjecture remains open, but has been proved for some classes of digraphs. We prove the conjecture for path-spine digraphs, defined as follows. A digraph D is path-spine if there exists a partition {X, Y} of V(D) such that D[X] has a Hamilton path and every arc in D[Y] belongs to a single path Q.

Resistance distance in directed cactus graphs

Let $G=(V,E)$ be a strongly connected and balanced digraph with vertex set $V=\{1,\dotsc,n\}$. The classical distance $d_{ij}$ between any two vertices $i$ and $j$ in $G$ is the minimum length of all the directed paths joining $i$ and $j$. The resistance distance (or, simply the resistance) between any two vertices $i$ and $j$ in $V$ is defined by $r_{ij}:=l_{ii}^{\dagger}+l_{jj}^{\dagger}-2l_{ij}^{\dagger}$, where $l_{pq}^{\dagger}$ is the $(p,q)^{\rm th}$ entry of the Moore-Penrose inverse of $L$ which is the Laplacian matrix of $G$. In practice, the resistance $r_{ij}$ is more significant than the classical distance. One reason for this is, numerical examples show that the resistance distance between $i$ and $j$ is always less than or equal to the classical distance, i.e., $r_{ij} \leq d_{ij}$. However, no proof for this inequality is known. In this paper, it is shown that this inequality holds for all directed cactus graphs.