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The connectivity of acyclic orientation graphs

Carla D. Savage, Cun-Quan Zhang
1998 Discrete Mathematics
The acyclic orientation graph, AO(G), of an undirected graph, G, is the graph whose vertices are the acyclic orientations of G and whose edges are the pairs of orientations differing only by the reversal  ...  We establish that if H is a triangle-free graph with minimum degree at least k, and the graph obtained by contracting the edges of a matching in H is k-connected, then H is k-connected.  ...  We are especially grateful to Paul Edelman for pointing out the relationship between acyclic orientations and polytopes, supplying the references, and leading us through the argument described in the appendix  ...

Notes on acyclic orientations and the shelling lemma

Komei Fukuda, Alain Prodon, Tadashi Sakuma
2001 Theoretical Computer Science
In this paper we study two lemmas on acyclic orientations and totally cyclic orientations of a graph, which can be derived from the shelling lemma in vector subspaces.  ...  We give simple graph theoretical proofs as well as a proof by the interpretations of the shelling lemma in the special setting of graphs.  ...  Connectivity of acyclic ip graph) . Let G be a connected graph with n nodes. Then the acyclic ip graph is (n − 1) connected; and has a vertex of degree (n − 1).  ...

Acyclic Orientation of Drawings

Eyal Ackerman, Kevin Buchin, Christian Knauer, Günter Rote
2010 Journal of Graph Algorithms and Applications
Given a set of curves in the plane or a topological graph, we ask for an orientation of the curves or edges which induces an acyclic orientation on the corresponding planar map.  ...  Depending on the maximum number of crossings on a curve or an edge, we provide algorithms and hardness proofs for this problem.  ...  We thank Michel Pocchiola for suggesting the problem concerning the acyclicity of curves in the plane. We also thank Scot Drysdale, Frank Hoffmann, and Klaus Kriegel for helpful discussions.  ...

Acyclic Orientation of Drawings [chapter]

Eyal Ackerman, Kevin Buchin, Christian Knauer, Günter Rote
2006 Lecture Notes in Computer Science
Given a set of curves in the plane or a topological graph, we ask for an orientation of the curves or edges which induces an acyclic orientation on the corresponding planar map.  ...  Depending on the maximum number of crossings on a curve or an edge, we provide algorithms and hardness proofs for this problem.  ...  We thank Michel Pocchiola for suggesting the problem concerning the acyclicity of curves in the plane. We also thank Scot Drysdale, Frank Hoffmann, and Klaus Kriegel for helpful discussions.  ...

Enumerating degree sequences in digraphs and a cycle–cocycle reversing system

Emeric Gioan
2007 European journal of combinatorics (Print)
We give some new enumerations of indegree sequences of orientations of a graph using the Tutte polynomial.  ...  In particular, concerning the cycle-cocycle reversing system, we show that its configurations are in bijection with indegree sequences of orientations having a given vertex (quasi-sink of the system) reachable  ...  The set of in-sequences of acyclic orientations of a graph G is in bijection with the set of acyclic orientations of G. Proof.  ...

Sampling and Counting Acyclic Orientations in Chordal Graphs (Student Abstract)

Wenbo Sun
2022 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
For a given undirected graph, an acyclic orientation is an assignment of directions to all of its edges which makes the resulting directed graph cycle-free.  ...  Counting of acyclic orientations of a given chordal graph can be done in polynomial time, but the previously known techniques do not seem to lead to a corresponding (efficient) sampler.  ...  Future Plans We view this work as a first step in the direction of counting and sampling different types of graph orientations on chordal graphs, such as bipolar orientations, sink-free orientations, and  ...

Every Orientation of a \$4\$-Chromatic Graph has a Non-Bipartite Acyclic Subgraph

Asaf Shapira
2022 Electronic Journal of Combinatorics
Let \$f(n)\$ denote the smallest integer such that every directed graph with chromatic number larger than \$f(n)\$ contains an acyclic subgraph with chromatic number larger than \$n\$.  ...  The problem of bounding this function was introduced by Addario-Berry et al., who noted that \$f(n) \leqslant n^2\$.  ...  Suppose G is a graph of chromatic number 4. Then every orientation of its edges contains an acyclic odd cycle.  ...

Full orientability of graphs with at most one dependent arc

Hsin-Hao Lai, Ko-Wei Lih, Li-Da Tong
2009 Discrete Applied Mathematics
Let d min (G) (d max (G)) denote the minimum (maximum) of the number of dependent arcs over all acyclic orientations of G.  ...  Suppose that D is an acyclic orientation of a graph G. An arc of D is dependent if its reversal creates a directed cycle.  ...  Acknowledgments The authors would like to thank the anonymous referees for their constructive comments.  ...

Acyclic orientations of complete bipartite graphs

Douglas B. West
1995 Discrete Mathematics
For a complete bipartite graph, the number of dependent edges in an acyclic orientation can be any integer from n-1 to e, where n and e are the number of vertices and edges in the graph.  ...  Acknowledgement The author thanks Paul Edelman for communicating the problem, Albert Li for interesting discussions about it, and a referee for suggesting stronger versions of the Lemmas.  ...  Every acyclic orientation of a connected simple graph G contains among its independent edges a spanning tree of G. Proof.  ...

Moving to Extremal Graph Parameters [article]

P.J. Cameron, C.A. Glass, R.U. Schumacher
2013 arXiv   pre-print
In this paper we develop a technique which provides answers for several different parameters: the numbers of edges in the line graph, acyclic orientations, cliques, and forests.  ...  Which graphs, in the class of all graphs with given numbers n and m of edges and vertices respectively, minimizes or maximizes the value of some graph parameter?  ...  Observe that an acyclic orientation of a graph defines an ordering on the vertices of the graph. For a clique this is a bijection.  ...

Page 8488 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews
Given a chordal bridgeless connected undirected graph G, the authors construct a linear time algorithm that finds a strongly connected orientation of G with oriented diameter at most one plus twice the  ...  During the last quarter century considerable work has followed on obtaining strongly connected orientations of connected bridgeless graphs that minimize the oriented diameter, particularly on special classes  ...

On the Broadcast Routing Problem in Computer Networks [article]

Brahim Chaourar
2020 arXiv   pre-print
Given an undirected graph G = (V, E), and a vertex r∈ V, an r-acyclic orientation of G is an orientation OE of the edges of G such that the digraph OG = (V, OE) is acyclic and r is the unique vertex with  ...  For w∈R^E_+, k(G, w) is the value of the w-maximum packing of r-arborescences for all r∈ V and all r-acyclic orientations OE of G.  ...  Future investigations can be studying the running time complexity of BRP and rBRP in general graphs and solve them in larger classes than outerplanar graphs.  ...

A Proof-checking Experiment on Representing Graphs as Membership Digraphs

Pierpaolo Calligaris, Eugenio G. Omodeo, Alexandru I. Tomescu
2013 Italian Conference on Computational Logic
These results will be enhanced in a forthcoming scenario, where every connected claw-free graph G will receive an extensional acyclic orientation and will, through such an orientation, be represented as  ...  We likewise proved that a graph whatsoever admits a weakly extensional and acyclic orientation; consequently, and in view of what precedes, one can regard its edges as membership arcs, each deprived of  ...  This eases things: for, any vertex with fewer than 2 incident edges in the spanning tree of a connected graph will be a non-cut vertex of the graph. Fig. 2 . 2 Fig. 2.  ...

Page 6198 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews
Summary: “Given a graph G, the acyclic orientation graph of G, denoted AO(G), is the graph whose vertices are the acyclic orientations of G, with two acyclic orientations joined by an edge in AO(G) iff  ...  We also give examples of graphs whose acyclic orientation graph is not Hamiltonian.  ...

The Prism of the Acyclic Orientation Graph is Hamiltonian

Gara Pruesse, Frank Ruskey
1995 Electronic Journal of Combinatorics
Every connected simple graph \$G\$ has an acyclic orientation.  ...  Define a graph \${AO}(G)\$ whose vertices are the acyclic orientations of \$G\$ and whose edges join orientations that differ by reversing the direction of a single edge.  ...  Introduction Every connected simple graph G has an acyclic orientation.  ...
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