DOCSLIB.ORG

One-Way Trail Orientations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
One-Way Trail Orientations One-Way Trail Orientations Anders Aamand1 University of Copenhagen, Copenhagen, Denmark [email protected] https://orcid.org/0000-0002-0402-0514 Niklas Hjuler2 University of Copenhagen, Copenhagen, Denmark [email protected] https://orcid.org/0000-0002-0815-670X Jacob Holm3 University of Copenhagen, Copenhagen, Denmark [email protected] https://orcid.org/0000-0001-6997-9251 Eva Rotenberg4 Technical University of Denmark, Lyngby, Denmark [email protected] https://orcid.org/0000-0001-5853-7909 Abstract Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins’ theorem [Robbins, Am. Math. Monthly, 1939] asserts that such an orientation exists if and only if the graph is 2-edge connected. A natural extension of this problem is the following: Suppose that the edges of the graph are partitioned into trails. Can the trails be oriented consistently such that the resulting directed graph is strongly connected? We show that 2-edge connectivity is again a sufficient condition and we provide a linear time algorithm for finding such an orientation. The generalised Robbins’ theorem [Boesch, Am. Math. Monthly, 1980] for mixed multigraphs asserts that the undirected edges of a mixed multigraph can be oriented to make the resulting directed graph strongly connected exactly when the mixed graph is strongly connected and the underlying graph is bridgeless. We consider the natural extension where the undirected edges of a mixed multigraph are partitioned into trails. It turns out that in this case the condition of the generalised Robbin’s Theorem is not sufficient. However, we show that as long as each cut either contains at least 2 undirected edges or directed edges in both directions, there exists an orientation of the trails such that the resulting directed graph is strongly connected. Moreover, if the condition is satisfied, we may start by orienting an arbitrary trail in an arbitrary direction. Using this result one obtains a very simple polynomial time algorithm for finding a strong trail orientation if it exists, both in the undirected and the mixed setting. 1 Research supported by Thorup’s Advanced Grant DFF-0602-02499B from the Danish Council for Independent Research and by his Investigator Grant 16582, Basic Algorithms Research Copenhagen (BARC), from the VILLUM Foundation. 2 This work is supported by the Innovation Fund Denmark through the DABAI project. 3 Research supported by Thorup’s Advanced Grant DFF-0602-02499B from the Danish Council for Independent Research and by his Investigator Grant 16582, Basic Algorithms Research Copenhagen (BARC), from the VILLUM Foundation. 4 This research was conducted during the fourth author’s time as a PhD student at University of Copenhagen. © Anders Aamand, Niklas Hjuler, Jacob Holm, and Eva Rotenberg; E A licensed under Creative Commons License CC-BY T C 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). S Editors: Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella; Article No. 6; pp. 6:1–6:13 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 6:2 One-Way Trail Orientations 2012 ACM Subject Classification Mathematics of computing → Graph algorithms, Mathemat- ics of computing → Paths and connectivity problems Keywords and phrases Graph algorithms, Robbins’ theorem, Graph orientation Digital Object Identifier 10.4230/LIPIcs.ICALP.2018.6 1 Introduction and motivation Suppose that the mayor of a small town decides to make all the streets one-way such that it is possible to get from any place to any other place without violating the orientations of the streets5. If all the streets are initially two-way then Robbins’ theorem [10] asserts that this can be done exactly when the corresponding graph is 2-edge connected. If, on the other hand some of the streets were already one-way in the beginning then the generalised Robbins’ theorem by Boesch [1] states that it can be done exactly when the corresponding “mixed” graph is strongly connected and the underlying graph is bridgeless. However, the proofs of both of these results assume that every street of the city corresponds to exactly one edge in the graph. This assumption hardly holds in any city in the world and therefore a more natural assumption is that every street corresponds to a trail (informally, a potentially self-crossing path) in the graph and that the edges of each trail must be oriented consistently6. In this paper we consider such graphs having their edges partitioned into trails. We prove that the trails can be oriented to make the resulting directed graph strongly connected exactly if the initial graph is 2-edge connected (note that this is precisely the condition of Robbins’ theorem). Not only do we show that the strong trail orientation problem in undirected 2-edge connected graphs always has a solution, we also provide a linear time algorithm for finding such an orientation. In doing so, we use an interesting combination of techniques that allow us to reduce to a graph with a number of 3-edge connected components that is linear in the number of edges. Using that the average size of these components is constant and that we can piece together solutions for the individual components we obtain an efficient algorithm. Finally, we will consider the generalised Robbins’ theorem in this new setting by allowing some edges to be oriented initially and supposing that the remaining edges are partitioned into trails. We will show that if each cut (V1,V2) in the graph has either at least 2 undirected edges going between V1 and V2 or at least 1 directed edge in each direction then it is possible to orient the trails making the resulting graph strongly connected. In fact, we show that if this condition is satisfied we may start by orienting an arbitrary trail in an arbitrary direction. Although this condition is not necessary it does give a simple algorithm for finding a trail orientation if it exists. Indeed, initially the graph may contain undirected edges that are forced in one direction by some cut. For finding a trail orientation if it exists we can thus orient forced trails in the forced direction. If there are no forced trails we orient any trail arbitrarily. 5 The motivation for doing so is that the streets of the town are very narrow and thus it is a great hassle when two cars unexpectedly meet. 6 This version of the problem was given to us through personal communication with Professor Robert E. Tarjan. A. Aamand, N. Hjuler, J. Holm, and E. Rotenberg 6:3 Figure 1 The graph is strongly connected and the underlying graph is 2-edge connected, but irrespective of the choice of orientation of the red trail, the graph will no longer be strongly connected. Note that in the mixed setting the feasibility depends on the trail decomposition which is not the case for the other results. That the condition from the generalised Robbins’ theorem is not sufficient can be seen from Figure 1. Earlier methods Several methods have already been applied for solving orientation problems in graphs where the goal is to make the resulting graph strongly connected. One approach used by Robbins [10] is to use that a 2-edge connected graph has an ear-decomposition. An ear decomposition of a graph is a partition of the set of edges into a cycle C and paths P1,...,Pt such that Pi has its two endpoints but none of its internal Si−1 vertices on C ∪ j=1 Pj . Given the existence of an ear decomposition of a 2-edge connected graph it is easy to prove Robbins’ theorem. Indeed, any choice of consistent orientations of the paths and the cycle gives a strongly connected graph. A second approach introduced by Tarjan [4] gives another simple proof of Robbins’ theorem. One can create a DFS tree in the graph rooted at a vertex v and orient all edges in the DFS tree away from v. The remaining edges are all back edges (see [4]) and are oriented towards v. It is easily verified that this gives a strong orientation if the graph is 2-edge connected. A similar approach was used by Chung et al. [2] in the context of the generalized Robbins’ theorem for mixed multigraphs. The above methods not only prove Robbins’ theorem, they also provide linear time algorithms for finding strong orientations of undirected or mixed multigraphs. However, none of the above methods have proven fruitful in our case. In case of the ear decomposition we would need one that is somehow compatible with the partitioning into trails, and this seems hard to guarantee. Similar problems appear when trying a DFS-approach. Neither does the proof by Boesch [1] of Robbins’ theorem for mixed multigraphs generalise to prove our result. Most importantly, the corresponding theorem would no longer be true for trail orientations as is shown by the example in Figure 1. Since the classical linear time algorithms rely on ear-decompositions and DFS searches, and since these approaches do not immediately work for trail partitions, our linear time algorithm will be a completely new approach to solving orientation problems. Structure of the paper The structure of this paper is as follows. In section 3 we prove our generalisation of Robbins’ theorem for undirected graphs partitioned into trails. In section 4 we study the case of mixed graphs. Finally in section 5 we provide our linear time algorithm for trail orientation in an undirected graph. I C A L P 2 0 1 8 6:4 One-Way Trail Orientations 2 Preliminaries Let us briefly review the concepts from graph theory that we will need.
Load more

Recommended publications
  • Strongly Connected Component
    Graph IV Ian Things that we would talk about ● DFS ● Tree ● Connectivity Useful website http://codeforces.com/blog/entry/16221 Recommended Practice Sites ● HKOJ ● Codeforces ● Topcoder ● Csacademy ● Atcoder ● USACO ● COCI Term in Directed Tree ● Consider node 4 – Node 2 is its parent – Node 1, 2 is its ancestors – Node 5 is its sibling – Node 6 is its child – Node 6, 7, 8 is its descendants ● Node 1 is the root DFS Forest ● When we do DFS on a graph, we would obtain a DFS forest. Noted that the graph is not necessarily a tree. ● Some of the information we get through the DFS is actually very useful, such as – Starting time of a node – Finishing time of a node – Parent of the node Some Tricks Using DFS Order ● Suppose vertex v is ancestor(not only parent) of u – Starting time of v < starting time of u – Finishing time of v > starting time of u ● st[v] < st[u] <= ft[u] < ft[v] ● O(1) to check if ancestor or not ● Flatten the tree to store subtree information(maybe using segment tree or other data structure to maintain) ● Super useful !!!!!!!!!! Partial Sum on Tree ● Given queries, each time increase all node from node v to node u by 1 ● Assume node v is ancestor of node u ● sum[u]++, sum[par[v]]-- ● Run dfs in root dfs(v) for all child u dfs(u) d[v] = d[v] + d[u] Types of Edges ● Tree edges – Edges that forms a tree ● Forward edges – Edges that go from a node to its descendants but itself is not a tree edge.
    [Show full text]
  • CLRS B.4 Graph Theory Definitions Unit 1: DFS Informally, a Graph
    CLRS B.4 Graph Theory Definitions Unit 1: DFS informally, a graph consists of “vertices” joined together by “edges,” e.g.,: example graph G0: 1 ···················•······························· ····························· ····························· ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ············· ···· ···· ·············· 2•···· ···· ···· ··· •· 3 ···· ···· ···· ···· ···· ··· ···· ···· ···· ···· ···· ···· ···· ···· ···· ···· ······· ······· ······· ······· ···· ···· ···· ··· ···· ···· ···· ···· ···· ··· ···· ···· ···· ···· ···· ···· ···· ···· ···· ···· ··············· ···· ···· ··············· 4•························· ···· ···· ························· • 5 ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ····························· ····························· ···················•································ 6 formally a graph is a pair (V, E) where V is a finite set of elements, called vertices E is a finite set of pairs of vertices, called edges if H is a graph, we can denote its vertex & edge sets as V (H) & E(H) respectively if the pairs of E are unordered, the graph is undirected if the pairs of E are ordered the graph is directed, or a digraph two vertices joined by an edge
    [Show full text]
  • Graph Theory
    1 Graph Theory “Begin at the beginning,” the King said, gravely, “and go on till you come to the end; then stop.” — Lewis Carroll, Alice in Wonderland The Pregolya River passes through a city once known as K¨onigsberg. In the 1700s seven bridges were situated across this river in a manner similar to what you see in Figure 1.1. The city’s residents enjoyed strolling on these bridges, but, as hard as they tried, no residentof the city was ever able to walk a route that crossed each of these bridges exactly once. The Swiss mathematician Leonhard Euler learned of this frustrating phenomenon, and in 1736 he wrote an article [98] about it. His work on the “K¨onigsberg Bridge Problem” is considered by many to be the beginning of the field of graph theory. FIGURE 1.1. The bridges in K¨onigsberg. J.M. Harris et al., Combinatorics and Graph Theory , DOI: 10.1007/978-0-387-79711-3 1, °c Springer Science+Business Media, LLC 2008 2 1. Graph Theory At first, the usefulness of Euler’s ideas and of “graph theory” itself was found only in solving puzzles and in analyzing games and other recreations. In the mid 1800s, however, people began to realize that graphs could be used to model many things that were of interest in society. For instance, the “Four Color Map Conjec- ture,” introduced by DeMorgan in 1852, was a famous problem that was seem- ingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions are colored differently.
    [Show full text]
  • Vizing's Theorem and Edge-Chromatic Graph
    VIZING'S THEOREM AND EDGE-CHROMATIC GRAPH THEORY ROBERT GREEN Abstract. This paper is an expository piece on edge-chromatic graph theory. The central theorem in this subject is that of Vizing. We shall then explore the properties of graphs where Vizing's upper bound on the chromatic index is tight, and graphs where the lower bound is tight. Finally, we will look at a few generalizations of Vizing's Theorem, as well as some related conjectures. Contents 1. Introduction & Some Basic Definitions 1 2. Vizing's Theorem 2 3. General Properties of Class One and Class Two Graphs 3 4. The Petersen Graph and Other Snarks 4 5. Generalizations and Conjectures Regarding Vizing's Theorem 6 Acknowledgments 8 References 8 1. Introduction & Some Basic Definitions Definition 1.1. An edge colouring of a graph G = (V; E) is a map C : E ! S, where S is a set of colours, such that for all e; f 2 E, if e and f share a vertex, then C(e) 6= C(f). Definition 1.2. The chromatic index of a graph χ0(G) is the minimum number of colours needed for a proper colouring of G. Definition 1.3. The degree of a vertex v, denoted by d(v), is the number of edges of G which have v as a vertex. The maximum degree of a graph is denoted by ∆(G) and the minimum degree of a graph is denoted by δ(G). Vizing's Theorem is the central theorem of edge-chromatic graph theory, since it provides an upper and lower bound for the chromatic index χ0(G) of any graph G.
    [Show full text]
  • Superposition and Constructions of Graphs Without Nowhere-Zero K-flows
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Europ. J. Combinatorics (2002) 23, 281–306 doi:10.1006/eujc.2001.0563 Available online at http://www.idealibrary.com on Superposition and Constructions of Graphs Without Nowhere-zero k-flows M ARTIN KOCHOL Using multi-terminal networks we build methods on constructing graphs without nowhere-zero group- and integer-valued flows. In this way we unify known constructions of snarks (nontrivial cubic graphs without edge-3-colorings, or equivalently, without nowhere-zero 4-flows) and provide new ones in the same process. Our methods also imply new complexity results about nowhere-zero flows in graphs and state equivalences of Tutte’s 3- and 5-flow conjectures with formally weaker statements. c 2002 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION Nowhere-zero flows in graphs have been introduced by Tutte [38–40]. Primarily he showed that a planar graph is face-k-colorable if and only if it admits a nowhere-zero k-flow (its edges can be oriented and assigned values ±1,..., ±(k − 1) so that the sum of the incoming values equals the sum of the outcoming ones for every vertex of the graph). Tutte also proved the classical equivalence result that a graph admits a nowhere-zero k-flow if and only if it admits a flow whose values are the nonzero elements of a finite abelian group of order k. Seymour [35] has proved that every bridgeless graph admits a nowhere-zero 6-flow, thereby improving the 8-flow theorem of Jaeger [16] and Kilpatrick [20].
    [Show full text]
  • Finding Articulation Points and Bridges Articulation Points Articulation Point
    Finding Articulation Points and Bridges Articulation Points Articulation Point Articulation Point A vertex v is an articulation point (also called cut vertex) if removing v increases the number of connected components. A graph with two articulation points. 3 / 1 Articulation Points Given I An undirected, connected graph G = (V; E) I A DFS-tree T with the root r Lemma A DFS on an undirected graph does not produce any cross edges. Conclusion I If a descendant u of a vertex v is adjacent to a vertex w, then w is a descendant or ancestor of v. 4 / 1 Removing a Vertex v Assume, we remove a vertex v 6= r from the graph. Case 1: v is an articulation point. I There is a descendant u of v which is no longer reachable from r. I Thus, there is no edge from the tree containing u to the tree containing r. Case 2: v is not an articulation point. I All descendants of v are still reachable from r. I Thus, for each descendant u, there is an edge connecting the tree containing u with the tree containing r. 5 / 1 Removing a Vertex v Problem I v might have multiple subtrees, some adjacent to ancestors of v, and some not adjacent. Observation I A subtree is not split further (we only remove v). Theorem A vertex v is articulation point if and only if v has a child u such that neither u nor any of u's descendants are adjacent to an ancestor of v. Question I How do we determine this efficiently for all vertices? 6 / 1 Detecting Descendant-Ancestor Adjacency Lowpoint The lowpoint low(v) of a vertex v is the lowest depth of a vertex which is adjacent to v or a descendant of v.
    [Show full text]
  • Strong Subgraph Connectivity of Digraphs: a Survey
    Strong Subgraph Connectivity of Digraphs: A Survey Yuefang Sun1 and Gregory Gutin2 1 Department of Mathematics, Shaoxing University Zhejiang 312000, P. R. China, [email protected] 2 School of Computer Science and Mathematics Royal Holloway, University of London Egham, Surrey, TW20 0EX, UK, [email protected] Abstract In this survey we overview known results on the strong subgraph k- connectivity and strong subgraph k-arc-connectivity of digraphs. After an introductory section, the paper is divided into four sections: basic results, algorithms and complexity, sharp bounds for strong subgraph k-(arc-)connectivity, minimally strong subgraph (k,ℓ)-(arc-) connected digraphs. This survey contains several conjectures and open problems for further study. Keywords: Strong subgraph k-connectivity; Strong subgraph k-arc- connectivity; Subdigraph packing; Directed q-linkage; Directed weak q-linkage; Semicomplete digraphs; Symmetric digraphs; Generalized k- connectivity; Generalized k-edge-connectivity. AMS subject classification (2010): 05C20, 05C35, 05C40, 05C70, 05C75, 05C76, 05C85, 68Q25, 68R10. 1 Introduction arXiv:1808.02740v1 [cs.DM] 8 Aug 2018 The generalized k-connectivity κk(G) of a graph G = (V, E) was intro- duced by Hager [14] in 1985 (2 ≤ k ≤ |V |). For a graph G = (V, E) and a set S ⊆ V of at least two vertices, an S-Steiner tree or, simply, an S-tree is a subgraph T of G which is a tree with S ⊆ V (T ). Two S-trees T1 and T2 are said to be edge-disjoint if E(T1) ∩ E(T2)= ∅. Two edge-disjoint S-trees T1 and T2 are said to be internally disjoint if V (T1) ∩ V (T2)= S.
    [Show full text]
  • Shattering, Graph Orientations, and Connectivity
    Shattering, Graph Orientations, and Connectivity [Extended Abstract] László Kozma Shay Moran Saarland University Max Planck Institute for Informatics Saarbrücken, Germany Saarbrücken, Germany [email protected] [email protected] ABSTRACT ing orientations with forbidden subgraphs. These problems We present a connection between two seemingly disparate have close ties with the theory of random graphs. An exam- fields: VC-theory and graph theory. This connection yields ple of work of this type is the paper of Alon and Yuster [4], natural correspondences between fundamental concepts in concerned with the number of orientations that do not con- VC-theory, such as shattering and VC-dimension, and well- tain a copy of a fixed tournament. studied concepts of graph theory related to connectivity, combinatorial optimization, forbidden subgraphs, and oth- In this paper we prove statements concerning properties of ers. orientations and subgraphs of a given graph by introducing a connection between VC-theory and graph theory. The In one direction, we use this connection to derive results following are a few examples of the statements that we prove: in graph theory. Our main tool is a generalization of the Sauer-Shelah Lemma [30, 10, 12]. Using this tool we obtain 1. Let G be a graph. Then: a series of inequalities and equalities related to properties of orientations of a graph. Some of these results appear to be the number the number the number of connected of strong of 2-edge- new, for others we give new and simple proofs. ≥ ≥ subgraphs orientations connected In the other direction, we present new illustrative examples of G of G subgraphs of G.
    [Show full text]
  • Incremental 2-Edge-Connectivity in Directed Graphs
    Incremental 2-Edge-Connectivity In Directed Graphs Loukas Georgiadis Giuseppe F. Italiano Nikos Parotsidis University of Ioannina University of Rome University of Rome Greece Tor Vergata Tor Vergata Italy Italy ICALP 2016, Rome Outline Definitions • 2-edge-connectivity in undirected graphs • 2-edge-connectivity in directed graphs • Problem definition • Known algorithm and our result High-level idea Basic ingredients • Dominators • Auxiliary components Tools Conclusion ICALP 2016, Rome Outline Definitions • 2-edge-connectivity in undirected graphs • 2-edge-connectivity in directed graphs • Problem definition • Known algorithm and our result High-level idea Basic ingredients • Dominators • Auxiliary components Tools Conclusion ICALP 2016, Rome Undirected: Connected components Let 퐺 = (푉, 퐸) be a undirected graph. • 퐺 is connected if there is a path between any two vertices. • The connected components of 퐺 are its maximal connected subgraphs. ICALP 2016, Rome Undirected: Connected components Let 퐺 = (푉, 퐸) be a connected undirected graph. • An edge is a bridge, if its removal increases the number of connected components. ICALP 2016, Rome Undirected: Connected components Let 퐺 = (푉, 퐸) be a connected undirected graph. • An edge is a bridge, if its removal increases the number of connected components. ICALP 2016, Rome Undirected: Connected components By Menger’s theorem, two vertices are 2-edge- connected iff the removal of any bridge leaves them in the same connected component. ICALP 2016, Rome Undirected: Connected components By Menger’s theorem, two vertices are 2-edge- connected iff the removal of any bridge leaves them in the same connected component. ICALP 2016, Rome Undirected: Connected components By Menger’s theorem, two vertices are 2-edge- connected iff the removal of any bridge leaves them in the same connected component.
    [Show full text]
  • Graph Theory (Connected Graphs)
    Graph Theory (Connected Graphs) Dr. Shubh N. Singh Department of Mathematics Central University of South Bihar May 14, 2020 2 Lecture-01: 0.1 Walks, Paths, and Cycles Let G = (V; E). If we think of the vertices of G as locations and the edges of G as roads between certain pairs of locations, then the graph G can be considered as modeling some community. There is a variety of kinds of trips that can be taken in the community. Let's start at some vertex u of a graph G. If we proceed from u to a neighbor v of u and then to a neighbor of v and so on, until we finally come to a stop at a vertex w, then we have just described a walk from u to v in G. More formally, Definition 0.1.1. Let G = (V; E), and let u; v 2 V .A walk from u to v (in short, a u − v walk) in G is a nonempty finite sequence/list u = v0; v1; : : : ; vk = v of vertices in G, where k ≥ 0 and fvi; vi+1g 2 E for all i = 0; 1; 2; : : : ; k − 1. If the vertices in a walk are all distinct, we call it a path in G. Note that all vertices or edges lie on (or, belong to ) a u − v walk need not be distinct; in fact u and v are not required to be distinct. Provided we continue to proceed from a vertex to one of its neighbors (and eventually stop), there is essentially no condi- tions on a walk in a graph.
    [Show full text]
  • One-Way Trail Orientations
    One-wasy trail orientation Aamand, Anders; Hjuler, Niklas; Holm, Jacob; Rotenberg, Eva Published in: 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 DOI: 10.4230/LIPIcs.ICALP.2018.6 Publication date: 2018 Document version Publisher's PDF, also known as Version of record Document license: CC BY Citation for published version (APA): Aamand, A., Hjuler, N., Holm, J., & Rotenberg, E. (2018). One-wasy trail orientation. In C. Kaklamanis, D. Marx, I. Chatzigiannakis, & D. Sannella (Eds.), 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 [6] Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik. Leibniz International Proceedings in Informatics, LIPIcs Vol. 107 https://doi.org/10.4230/LIPIcs.ICALP.2018.6 Download date: 26. Sep. 2021 One-Way Trail Orientations Anders Aamand1 University of Copenhagen, Copenhagen, Denmark [email protected] https://orcid.org/0000-0002-0402-0514 Niklas Hjuler2 University of Copenhagen, Copenhagen, Denmark [email protected] https://orcid.org/0000-0002-0815-670X Jacob Holm3 University of Copenhagen, Copenhagen, Denmark [email protected] https://orcid.org/0000-0001-6997-9251 Eva Rotenberg4 Technical University of Denmark, Lyngby, Denmark [email protected] https://orcid.org/0000-0001-5853-7909 Abstract Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected? Robbins’ theorem [Robbins, Am. Math. Monthly, 1939] asserts that such an orientation exists if and only if the graph is 2-edge connected. A natural extension of this problem is the following: Suppose that the edges of the graph are partitioned into trails. Can the trails be oriented consistently such that the resulting directed graph is strongly connected? We show that 2-edge connectivity is again a sufficient condition and we provide a linear time algorithm for finding such an orientation.
    [Show full text]
  • NOWHERE-ZERO K-FLOWS on GRAPHS Let G = (V,E) Be a Graph
    NOWHERE-ZERO ~k-FLOWS ON GRAPHS MATTHIAS BECK, ALYSSA CUYJET, GORDON ROJAS KIRBY, MOLLY STUBBLEFIELD, AND MICHAEL YOUNG Abstract. We introduce and study a multivariate function that counts nowhere-zero flows on a graph G, in which each edge of G has an individual capacity. We prove that the associated counting function is a piecewise-defined polynomial in these capacities, which satisfy a combinatorial reciprocity law that incorporates totally cyclic orientations of G. Let G = (V; E) be a graph, which we allow to have multiple edges. Fix an orientation of G, i.e., for each edge e = uv 2 E we assign one of u and v to be the head h(e) and the other to be the tail t(e) of e.A flow on (this orientation of) G is a labeling x 2 AE of the edges of G with values in an Abelian group A such that for every node v 2 V , X X xe = xe ; h(e)=v t(e)=v i.e., we have conservation of flow at each node. We are interested in counting such flows that are nowhere zero, i.e., xe 6= 0 for every edge e 2 E. The case A = Zk goes back to Tutte [11], who proved that the number '(k) of nowhere-zero Zk-flows is a polynomial in k. (Tutte introduced this counting function as a dual concept to the chromatic polynomial.) In the case A = Z, a k-flow takes on integer values in {−k + 1; : : : ; k − 1g. The fact that the number '(k) of nowhere-zero k-flows is also a polynomial in k is due to Kochol [7].
    [Show full text]