An acyclic set in a digraph is a set of vertices that induces an acyclic
subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$
vertices without directed 2-cycles possesses an acyclic set of size at least
$3n/5$. We prove this conjecture for digraphs where every directed cycle has
length at least 8. More generally, if $g$ is the length of the shortest
directed cycle, we show that there exists an acyclic set of size at least $(1 -
3/g)n$.Comment: 9 page

Golowich, Noah

Rolnick, David

The Electronic Journal of Combinatorics

English

arXiv

An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on n vertices without directed 2-cycles possesses an acyclic set of size at least 3n/5. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if g is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least (1 − 3/g)n. 1

Noah Golowich

David Rolnick

CiteSeerX

Acyclic Subgraphs of Planar Digraphs

An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on n vertices without directed 2-cycles possesses an acyclic set of size at least 3n=5. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if g is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least (1 - 3/g)n

Rolnick, David S.

DSpace@MIT

Acyclic Subgraphs of Planar DigraphsNoah GolowichResearch Science InstituteDepartment of MathematicsMassachusetts Institute of TechnologyCambridge, Massachusetts, U.S.A.ngolowich@college.harvard.eduDavid RolnickDepartment of MathematicsMassachusetts Institute of TechnologyCambridge, Massachusetts, U.S.A.drolnick@mit.eduSubmitted: Aug 11, 2014; Accepted: Jun 18, 2015; Published: Jul 1, 2015Mathematics Subject Classifications: 05C10, 05C15AbstractAn acyclic set in a digraph is a set of vertices that induces an acyclic subgraph.In 2011, Harutyunyan conjectured that every planar digraph on n vertices withoutdirected 2-cycles possesses an acyclic set of size at least 3n/5. We prove this conjec-ture for digraphs where every directed cycle has length at least 8. More generally,if g is the length of the shortest directed cycle, we show that there exists an acyclicset of size at least (1− 3/g)n.Keywords: acyclic set, chromatic number, digraph, planar graph, directed cut1 IntroductionA proper vertex coloring of an undirected graph G partitions the vertices into independentsets. It is natural to try to reformulate this notion for directed graphs (digraphs). Anacyclic set in a digraph is a set of vertices whose induced subgraph contains no directedcycle. The digraph chromatic number of a digraph D, denoted χA(D), is the minimalnumber of acyclic sets into which the vertices of D may be partitioned. In this paper, weconsider oriented graphs, which are digraphs such that at most one arc connects any pairof vertices.Although recent results [1, 3, 8, 10] suggest that the digraph chromatic number be-haves similarly to the undirected chromatic number, much still remains to be learned.For instance, Neumann-Lara conjectured that all oriented planar graphs have digraphchromatic number at most 2.Conjecture 1 ([12]). Every oriented planar graph is digraph 2-colorable.the electronic journal of combinatorics 22(3) (2015), #P3.7 1The fact that Conjecture 1 holds if one allows three colors instead of two is well-knownand is an immediate consequence of 5-degeneracy of planar graphs (see, for instance [3]).One approach to Conjecture 1 has been to look for lower bounds on the size of the largestacyclic set of a planar oriented graph. Borodin [4] showed that in any undirected planargraph there exists an induced forest of size 2n/5, where n denotes the number of verticesof the graph. This result implies that there must exist an acyclic set of size at least 2n/5in any oriented planar graph. Albertson and Berman [2] moreover conjectured that everyundirected planar graph has an induced forest of size at least n/2, which would implythat every oriented planar graph has an acyclic set of size n/2. Harutyunyan and Mohar[7] recently conjectured an even stronger bound on the maximum size of an acyclic set,which, if true, would be tight:Conjecture 2 ([7]). Every oriented planar graph on n vertices contains an acyclic set ofsize at least 3n/5.Finding the largest size of an acyclic set in a digraph is equivalent to finding a set ofvertices of minimum size which has a non-empty intersection with each directed cycle. In[9], Jain et al. investigated the applications of this problem in deadlock resolution.The digirth of a directed graph is the length of its shortest directed cycle. Fox andPach [5] considered how many vertices must be removed from an undirected graph Gto yield a forest, where G belongs to a hereditary family of graphs with small separators(such as the family of planar graphs). Their approach also applies to the directed case andshows that in an oriented planar graph there is always an acyclic set of size (1− c/√g)n,where c is an absolute constant and g is the digirth.Recently, Harutyunyan and Mohar [7] proved that every oriented planar graph ofdigirth at least 5 is digraph 2-colorable, and therefore contains an acyclic set of size atleast n/2. They used an intricate vertex-discharging method to show that any minimalcounterexample must contain at least one of 25 specific configurations of vertices and arcs.The authors posed the problem of finding a simpler approach for considering digraphcolorings in oriented planar graphs.In this paper, we introduce such an approach to prove Theorem 7, which improvesknown lower bounds on the largest acyclic set in oriented planar graphs of digirth at least4. Specifically, we give a short proof of the fact that every oriented planar graph on nvertices and of digirth g possesses an acyclic set of size at least (1− 3/g)n, thus provingConjecture 2 when g > 8. We also give a slightly stronger bound for the cases g = 4 andg = 5. We prove Theorem 7 by using a corollary of the Lucchesi-Younger theorem [11] tofind an upper bound on the size of the minimum feedback arc set of an oriented planargraph.In Section 2, we prove our main result, Theorem 7. In Section 3, we describe somepotential extensions of Theorem 7 and difficulties that arise.the electronic journal of combinatorics 22(3) (2015), #P3.7 22 Bounds on the largest acyclic setIn this section, we use a corollary of the Lucchesi-Younger theorem to prove Theorem 7.We begin with some definitions. Given a directed graph D and a subset X of its verticesV (D), we define X¯ = V (D)\X. If every arc between X and X¯ is directed from X toX¯, then the set of such arcs is called a directed cut. A dijoin is a set of arcs that has anon-empty intersection with every directed cut.The Lucchesi-Younger theorem [11] gives the minimum size of a dijoin of a digraph:Theorem 3 (Lucchesi-Younger [11]). The minimum cardinality of a dijoin in a directedgraph D is equal to the maximum number of pairwise disjoint directed cuts of D.In the case that D is planar, the Lucchesi-Younger theorem has a useful corollary forthe dual of D. Given an oriented planar graph D, the dual of D, denoted D?, is defined asfollows. For a given planar embedding of D, construct a vertex of D? within each face ofD. For each arc uv of D separating faces f and g of D, a corresponding arc f ?g? ∈ A(D?)is drawn between vertices f ? and g?. The direction of arc f ?g? is defined so that as itcrosses uv, v is on the left. It is simple to verify that the graph D? does not depend onthe planar embedding of D.The following well-known result establishes a bijection between the directed cycles ofan oriented planar graph and the directed cuts in its dual:Proposition 4. If D is an oriented planar graph, then the directed cycles of D are inone-to-one correspondence with the directed cuts in D?.Proof. Pick a planar embedding of D and then embed D? in the plane as defined above.Given a directed cycle C in D, notice that all arcs of D? crossing an arc of C must travelin the same direction: specifically, if C is oriented clockwise, then all arcs of D? crossingC point inwards, and if C is oriented counterclockwise, then the arcs of D? crossing Cpoint outwards. Let X ⊂ V (D?) consist of the vertices of D? corresponding to all facesof D inside C, and therefore the arcs of D? connecting X and V (D?)\X form a directedcut.Conversely, given a directed cut of V (D?), we reverse the method above to obtain adirected cycle of D.Given a directed graph, a feedback arc set is a set of arcs, the removal of whicheliminates all directed cycles. We will be concerned with minimum feedback arc sets,namely those of minimum cardinality. Proposition 4 establishes the following corollary ofthe Lucchesi-Younger theorem:Corollary 5 ([11]). For an oriented planar graph, the minimum size of a feedback arc setis equal to the maximum number of arc-disjoint directed cycles.Proof. Given an oriented planar graph D, by the Lucchesi-Younger theorem, the minimumcardinality of a dijoin of D? is equal to the maximum number of disjoint directed cuts ofD?. Now, by Proposition 4, any dijoin of D? corresponds to a unique feedback arc set ofD of the same size, and any set of disjoint directed cuts of D? corresponds to a unique setof arc-disjoint directed cycles of D, also of the same size. This completes the proof.the electronic journal of combinatorics 22(3) (2015), #P3.7 3We also need one well-known lemma, which follows easily from Euler’s formula forplanar graphs.Lemma 6. Any planar graph G with n vertices and m arcs satisfies m 6 3n− 6.We now are ready to state and prove our main theorem.Theorem 7. If D is an oriented planar graph with digirth g on n vertices, then thereexists an acyclic set in D of size at least n − 3n/g. Moreover, if g = 4, there exists anacyclic set of size at least 5n/12, and if g = 5, there exists an acyclic set of size at least7n/15.Proof. A vertex cover of a given arc set A is a set S of vertices for which every arc in A isincident to some vertex in S. Observe that if S is a vertex cover of a feedback arc set ina digraph D, then the complement of S in V (D) is an acyclic set. This is because everydirected cycle must include an element of the feedback arc set.Given an oriented planar graph D of digirth g, let H be a collection of arc-disjointdirected cycles, each of which must have length at least g. Thus, by Lemma 6 the numberof cycles in H is at most a(D)/g 6 3n/g, where a(D) denotes the number of arcs of D.Corollary 5 implies that there exists a feedback arc set with cardinality at most 3n/g.Removing a vertex cover of this feedback arc set, we are left with an acyclic set of at leastn− 3n/g vertices.For the cases g = 4, 5, we now derive a better bound by using the greedy algorithm toobtain a smaller vertex cover of the feedback arc set. This is possible because 3n/g > n/2,meaning that some vertices are incident to 2 or more feedback arcs. Suppose that ourfeedback arc set consists initially of f arcs, where f 6 3n/g.Let d equal the number of feedback arcs minus half the number of vertices. Thus,initially d = f − n/2. At each step, we remove the vertex v that is incident to the mostfeedback arcs, together with all feedback arcs incident to v. As long as d > 0, each stepremoves one vertex and at least two feedback arcs. Such a step decreases d by at least3/2. Let m be the number of steps taken before d 6 0, and let d′ 6 0 be the final valueof d. We conclude thatm 6 f − n/2− d′3/2=2f − n− 2d′3.After m steps, the number of vertices remaining is n−m, so the number of feedbackarcs remaining is d′ + (n−m)/2. We remove one vertex from each of these feedback arcsso that the vertices remaining at the end form an acyclic set. The total number of verticesremoved ism+ d′ +n−m26 n2+ d′ +12· 2f − n− 2d′3=n+ f + 2d′36 n+ f36 n3+ng.the electronic journal of combinatorics 22(3) (2015), #P3.7 4The number of vertices remaining in our acyclic set is thus at least2n3− ng.This is equal to 5n/12 for g = 4 and 7n/15 for g = 5.Note that our result for g = 5 and g = 6 is already known; in fact, as mentionedbefore, the main result of [7] implies the existence of an acyclic set of size at least n/2for g > 5. However, our bound of 5n/12 improves all previous results for g = 4 and ourbound of n− 3n/g improves all previous results for g > 7.3 Further directionsIn this section we describe methods that might be used to strengthen our main result,and difficulties that arise.3.1 Digraph 2-coloringsHarutyunyan and Mohar [7] asked whether there is a simple proof of the fact that orientedplanar graphs of large digirth are digraph 2-colorable. We showed above that we can finda very large acyclic set in an oriented planar graph of large digirth. It is natural to askwhether, given a minimum feedback arc set as in Theorem 7, we can choose a vertex coverY of the feedback arc set such that Y induces an acyclic subgraph. This would implythat oriented planar graphs of large digirth are digraph 2-colorable, because the subgraphinduced by V (D)\Y contains no feedback arcs and hence is acyclic. We now show thatthere are oriented planar graphs for which no such Y can be found.Proposition 8. There exists an oriented planar graph D such that, for any minimumfeedback arc set F of D, and for any vertex cover Y of F , there is a directed cycle in thesubgraph induced by Y .Proof. We show that the digraph D in Figure 1 satisfies the necessary conditions. Theunique minimum feedback arc set {ab, bc, ac, ed, gf, hi} is indicated with dashed lines.Notice that any vertex cover of the minimum feedback arc set must contain at least 2vertices from the central triangle abc, suppose a and b without loss of generality. We mustnow choose either d or e to cover arc de. However, if we choose d, then abd is a directed3-cycle, and if we choose e, then abe is a directed 3-cycle. This completes the proof.3.2 Improving bounds for small gFor small g, the bound we obtain in Theorem 7 on the maximum acyclic set of an orientedplanar graph on n vertices is less than n/2, and it is natural to ask whether we can improveupon our bound. In Propositions 9 and 10, we now present two classes of planar orientedgraphs, of digirth 3 and 4, respectively, for which the minimum size of a set of feedbackthe electronic journal of combinatorics 22(3) (2015), #P3.7 5a bcd efghijklmnopqrstuFigure 1: The oriented planar graph D. The unique minimum feedback arc set of D isshown in dashed lines. Note that D is planar since arc ea can be relocated around abeqd,and similarly for gb and ci.arcs is significantly greater than n/2. Therefore, removing one vertex from each feedbackarc yields an acyclic set of size less than n/2. Although some feedback arcs may overlapin certain vertices, as we demonstrate in the proof of Theorem 7, applying the greedyalgorithm still does not show the existence of an acyclic set of size at least n/2. Hencethese classes of graphs suggest that the methods used in Theorem 7 are unlikely to provesignificantly stronger bounds for digirth 3 and 4.Given a digraph D, let f(D) denote the size of a minimum set of feedback arcs.Proposition 9. There exists an infinite family of digraphs D3, such that for all D ∈ D3:• D is planar, with digirth 3.• f(D) = |V (D)| − 2.Proof. We define Di ∈ D3 recursively. Let D0 be a directed 3-cycle, and for i > 0,construct Di+1 as follows from a planar embedding of Di. Pick some face abc of Di thatforms a directed 3-cycle. Construct vertices d, e, f within abc, with arcs as shown inFigure 2.the electronic journal of combinatorics 22(3) (2015), #P3.7 6Observe that |V (Di+1)| = |V (Di)|+3 and f(Di+1) = f(Di)+3. Since |V (D0)| = 3 andf(D0) = 1, it follows by induction that f(Di) = |V (Di)| − 2 for every i. By construction,Di is planar and has digirth 3.abcde fFigure 2: Directed octahedral pattern for generating graphs of digirth 3 with large mini-mum feedback arc set.Proposition 10. There exists an infinite family of digraphs D4, such that for all D ∈ D4:• D is planar, with digirth 4.• f(D) = 58|V (D)| − 32.Proof. As with D3, we define Di ∈ D4 recursively. Let D0 be a directed 4-cycle, and fori > 0, construct Di+1 as follows from a planar embedding of Di. Pick some face abcd ofDi that forms a directed 4-cycle. Construct vertices e, f, g, h, i, j, k, l within abcd, witharcs as shown in Figure 3.Observe that |V (Di+1)| = |V (Di)|+8 and f(Di+1) = f(Di)+5. Since |V (D0)| = 4 andf(D0) = 1, it follows by induction that f(Di) =58|V (Di)|− 32 for every i. By construction,Di is planar and has digirth 4.We have defined a class of digraphs D3 by recursive addition of an octahedral pattern,and a class D4 by recursive addition of a cuboctahedral pattern. It is possible to definea similar class D5, containing planar digraphs of digirth 5, by recursive addition of anicosidodecahedral pattern. However, the resulting relation on f(D) falls short of |V (D)|/2:f(D) =1125|V (D)| − 65.Thus, for D ∈ D5, removing a vertex from each of a minimum set of feedback arcs yieldsan acyclic set of size greater than |V (D)|/2. We cannot say whether the methods ofTheorem 7 may hold for general planar graphs of digirth 5.the electronic journal of combinatorics 22(3) (2015), #P3.7 7a bcdefghi jklFigure 3: Directed cuboctahedral pattern for generating graphs of digirth 4 with largeminimum feedback arc set.AcknowledgementsThe authors would like to thank Jacob Fox for calling our attention to this problem andTanya Khovanova for helpful discussions and review. We would also like to thank the MITDepartment of Mathematics, the Center for Excellence in Education, and the ResearchScience Institute for their support of this research.References[1] R. Aharoni, E. Berger, and O. Kfir. Acyclic systems of representatives and acycliccolorings of digraphs. Journal of Graph Theory, pages 177–189, 2008.[2] M. O. Albertson and D. M. Berman. A conjecture on planar graphs. Graph Theoryand Related Topics (J.A. Bondy and U.S.R. Murty, eds.), page 357. Academic Press,1979.[3] D. Bokal, G. Fijavzˇ, M. Juvan, P. Mark Kayll, and B. Mohar. The circular chromaticnumber of a digraph. Journal of Graph Theory, 46(3): 227–240, 2004.[4] O. V. Borodin. On acyclic colorings of planar graphs. Discrete Mathematics, 25: 211–236, 1979.[5] J. Fox and J. Pach. A separator theorem for string graphs and its applications.Combinatorics, Probability and Computing, 19(3): 371–390, 2010.the electronic journal of combinatorics 22(3) (2015), #P3.7 8[6] A. Harutyunyan. Brooks-type results for coloring of digraphs. PhD thesis, Burnaby,British Columbia, 2011.[7] A. Harutyunyan and B. Mohar. Planar digraphs of digirth 5 are 2-colorable, 2014.arXiv:1401.2213[8] A. Harutyunyan and B. Mohar. Two results on the digraph chromatic number.Discrete Mathematics, 312(10): 1823–1826, 2012.[9] K. Jain, M. T. Hajiaghayi, and K. Talwar. The generalized deadlock resolutionproblem. 32nd International Colloquium on Automata, Languages and Programming,3580: 853–865, 2005.[10] P. Keevash, Z. Li, B. Mohar, and B. Reed. Digraph girth via chromatic number.SIAM Journal on Discrete Mathematics, 27(2): 693–696, 2013.[11] C. Lucchesi and D. H. Younger. A minimax theorem for directed graphs. Journal ofLondon Mathematical Society, 2: 369–374, 1978.[12] V. Neumann-Lara. Vertex colorings in digraphs. Some problems. Seminar notes,University of Waterloo, 1985.the electronic journal of combinatorics 22(3) (2015), #P3.7 9

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