#### Publikationen im Fachgebiet

Anzahl der Treffer: 196
Erstellt: Sat, 23 Sep 2023 23:10:13 +0200 in 0.0673 sec

Bang-Jensen, Jørgen; Hörsch, Florian; Kriesell, Matthias
Complexity of (arc)-connectivity problems involving arc-reversals or deorientations. - In: Theoretical computer science, Bd. 973 (2023), 114097

By a well known theorem of Robbins, a graph G has a strongly connected orientation if and only if G is 2-edge-connected and it is easy to find, in linear time, either a cut edge of G or a strong orientation of G. A result of Durand de Gevigney shows that for every it is NP-hard to decide if a given graph G has a k-strong orientation. Thomassen showed that one can check in polynomial time whether a given graph has a 2-strong orientation. This implies that for a given digraph D we can determine in polynomial time whether we can reorient (=reverse) some arcs of to obtain a 2-strong digraph. This naturally leads to the question of determining the minimum number of such arcs to reverse before the resulting graph is 2-strong. In this paper we show that finding this number is NP-hard. If a 2-connected graph G has no 2-strong orientation, we may ask how many of its edges we may orient so that the resulting mixed graph is still 2-strong. Similarly, we may ask for a 2-edge-connected graph G how many of its edges we can orient such that the resulting mixed graph remains 2-arc-strong. We prove that when restricted to graphs satisfying suitable connectivity conditions, both of these problems are equivalent to finding the minimum number of edges we must double in a 2-edge-connected graph in order to obtain a 4-edge-connected graph. Using this, we show that all these three problems are NP-hard. Finally, we consider the operation of deorienting an arc uv of a digraph D meaning replacing it by an undirected edge between the same vertices. In terms of connectivity properties, this is equivalent to adding the opposite arc vu to D. We prove that for every it is NP-hard to find the minimum number of arcs to deorient in a digraph D in order to obtain an ℓ-strong digraph.

https://doi.org/10.1016/j.tcs.2023.114097
Chan, Tsz Lung; Kriesell, Matthias; Schmidt, Jens M.
Contractible edges in longest cycles. - In: Journal of graph theory, ISSN 1097-0118, Bd. 103 (2023), 3, S. 542-563

https://doi.org/10.1002/jgt.22935
Hörsch, Florian;
Globally balancing spanning trees. - In: European journal of combinatorics, Bd. 109 (2023), 103644

https://doi.org/10.1016/j.ejc.2022.103644
Hörsch, Florian; Szigeti, Zoltán
On the complexity of finding well-balanced orientations with upper bounds on the out-degrees. - In: Journal of combinatorial optimization, ISSN 1573-2886, Bd. 45 (2023), 1, 30, S. 1-14

https://doi.org/10.1007/s10878-022-00962-y
Hörsch, Florian; Szigeti, Zoltán
Reachability in arborescence packings. - In: Discrete applied mathematics, ISSN 1872-6771, Bd. 320 (2022), S. 170-183

Fortier et al. proposed several research problems on packing arborescences and settled some of them. Others were later solved by Matsuoka and Tanigawa and by Gao and Yang. The last open problem is settled in this article. We show how to turn an inductive idea used in the latter two articles into a simple proof technique that allows to relate previous results on arborescence packings. We prove that a strong version of Edmonds’ theorem on packing spanning arborescences implies Kamiyama, Katoh and Takizawa’s result on packing reachability arborescences and that Durand de Gevigney, Nguyen and Szigeti’s theorem on matroid-based packing of arborescences implies Király’s result on matroid-reachability-based packing of arborescences. Further, we deduce a new result on matroid-reachability-based packing of mixed hyperarborescences from a theorem on matroid-based packing of mixed hyperarborescences due to Fortier et al.. Finally, we deal with the algorithmic aspects of the problems considered. We first obtain algorithms to find the desired packings of arborescences in all settings and then apply Edmonds’ weighted matroid intersection algorithm to also find solutions minimizing a given weight function.

https://doi.org/10.1016/j.dam.2022.05.018
Bang-Jensen, Jørgen; Kriesell, Matthias
Good acyclic orientations of 4-regular 4-connected graphs. - In: Journal of graph theory, ISSN 1097-0118, Bd. 100 (2022), 4, S. 698-720

An st-ordering of a graph G=(V,E) is an ordering v1,v2,…,vn of its vertex set such that s=v1,t=vn and every vertex vi with i=2,3,…,n-1 has both a lower numbered and a higher numbered neighbor. Such orderings have played an important role in algorithms for planarity testing. It is well-known that every 2-connected graph has an st-ordering for every choice of distinct vertices s,t. An st-ordering of a graph G corresponds directly to a so-called bipolar orientation of G, that is, an acyclic orientation D of G in which s is the unique source and t is the unique sink. Clearly every bipolar orientation of a graph has an out-branching rooted at the source vertex and an in-branching rooted at the sink vertex. In this paper, we study graphs which admit a bipolar orientation that contains an out-branching and in-branching which are arc-disjoint (such an orientation is called good). A 2T-graph is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. Clearly a graph has a good orientation if and only if it contains a spanning 2T-graph with a good orientation, implying that 2T-graphs play a central role. It is a well-known result due to Tutte and Nash-Williams, respectively, that every 4-edge-connected graph contains a spanning 2T-graph. Vertex-minimal 2T-graphs with at least two vertices, also known as generic circuits, play an important role in rigidity theory for graphs. Recently with Bessy and Huang we proved that every generic circuit has a good orientation. In fact, we may specify the roots of the two branchings arbitrarily as long as they are distinct. Using this, several results on good orientations of 2T-graphs were obtained. It is an open problem whether there exists a polynomial algorithm for deciding whether a given 2T-graph has a good orientation. Complex constructions of 2T-graphs with no good orientation were given in work by Bang-Jensen, Bessy, Huang and Kriesell (2021) indicating that the problem might be very difficult. In this paper, we focus on so-called quartics which are 2T-graphs where every vertex has degree 3 or 4. We identify a sufficient condition for a quartic to have a good orientation, give a polynomial algorithm to recognize quartics satisfying the condition and a polynomial algorithm to produce a good orientation when this condition is met. As a consequence of these results we prove that every 4-regular and 4-connected graph has a good orientation, where, as for generic circuits, we may specify the roots of the two branchings arbitrarily as long as they are distinct. We also provide evidence that even for quartics it may be difficult to find a characterization of those instances which have a good orientation. We also show that every graph on n≥8 vertices and of minimum degree at least has a good orientation. Finally we pose a number of open problems.

https://doi.org/10.1002/jgt.22803
Hörsch, Florian;
Checking the admissibility of odd-vertex pairings is hard. - In: Discrete applied mathematics, ISSN 1872-6771, Bd. 317 (2022), S. 42-48

Nash-Williams proved that every graph has a well-balanced orientation. A key ingredient in his proof is admissible odd-vertex pairings. We show that for two slightly different definitions of admissible odd-vertex pairings, deciding whether a given odd-vertex pairing is admissible is co-NP-complete. This resolves a question of Frank. We also show that deciding whether a given graph has an orientation that satisfies arbitrary local arc-connectivity requirements is NP-complete.

https://doi.org/10.1016/j.dam.2022.04.004
Bang-Jensen, Jørgen; Havet, Frederic; Kriesell, Matthias; Yeo, Anders
Low chromatic spanning sub(di)graphs with prescribed degree or connectivity properties. - In: Journal of graph theory, ISSN 1097-0118, Bd. 99 (2022), 4, S. 615-636

https://doi.org/10.1002/jgt.22755
Kriesell, Matthias;
A note on uniquely 10-colorable graphs. - In: Journal of graph theory, ISSN 1097-0118, Bd. 98 (2021), 1, S. 24-26

Hadwiger conjectured that every graph of chromatic number k admits a clique minor of order k. Here we prove for k ≤ 10, that every graph of chromatic number k with a unique k-coloring (up to the color names) admits a clique minor of order k. The proof does not rely on the Four Color Theorem.

https://doi.org/10.1002/jgt.22679
Bang-Jensen, Jørgen; Bessy, Stéphane; Huang, Jing; Kriesell, Matthias
Good orientations of unions of edge-disjoint spanning trees. - In: Journal of graph theory, ISSN 1097-0118, Bd. 96 (2021), 4, S. 594-618

In this paper, we exhibit connections between the following subjects: Tree packing in graphs and digraphs (both behave completely different), the rigidity matroid of a graph, Henneberg moves on trees, the conjectures of Thomassen and Matthews and Sumner, and (s,t)-orderings of digraphs. We do this by studying graphs which admit acyclic orientations that contain an out-branching and in-branching which are arc-disjoint (such an orientation is called good). A 2T-graph is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. It is a well-known result due to Tutte and Nash-Williams, respectively, that every 4-edge-connected graph contains a spanning 2T-graph. Vertex-minimal 2T-graphs with at least two vertices which are known as generic circuits play an important role in rigidity theory for graphs. We prove that every generic circuit has a good orientation. Using this result we prove that if G is 2T-graph whose vertex set has a partition V1,V2, ,Vk so that each Vi induces a generic circuit Gi of G and the set of edges between different Gi's form a matching in G, then G has a good orientation. We also obtain a characterization for the case when the set of edges between different Gi's form a double tree, that is, if we contract each Gi to one vertex, and delete parallel edges we obtain a tree. All our proofs are constructive and imply polynomial algorithms for finding the desired good orderings and the pairs of arc-disjoint branchings which certify that the orderings are good. We identify a structure which can be used to certify that a given 2T-graph does not have a good orientation.

https://doi.org/10.1002/jgt.22633