Halin graph

A Halin graph.

In graph theory, a Halin graph is a type of planar graph. It is constructed from a tree that has at least four vertices, none of which have exactly two neighbors. The tree is drawn in the plane so none of its edges cross; then edges are added that connect all its leaves into a cycle.[1] Halin graphs are named after German mathematician Rudolf Halin, who studied them in 1971,[2] but the cubic Halin graphs had already been studied over a century earlier by Kirkman.[3]

Construction

A triangular prism, constructed as a Halin graph from a six-vertex tree

A Halin graph is constructed as follows. Let T be a tree with more than three vertices, such that no vertex of T has degree two (that is, no vertex has exactly two neighbors), embedded in the plane. Then a Halin graph is constructed by adding to T a cycle through each of its leaves, such that the augmented graph remains planar.

Examples

Wheel graphs

A star is a tree with exactly one internal vertex. Applying the Halin graph construction to a star produces a wheel graph, the graph of a pyramid. The graph of a triangular prism is also a Halin graph: it can be drawn so that one of its rectangular faces is the exterior cycle, and the remaining edges form a tree with four leaves, two interior vertices, and five edges.

The Frucht graph, one of the two smallest cubic graphs with no nontrivial graph automorphisms, is also a Halin graph.

Properties

Every Halin graph is 3-connected, meaning that it is not possible to delete two vertices from it and disconnect the remaining vertices. It is edge-minimal 3-connected, meaning that if any one of its edges is removed, the remaining graph will no longer be 3-connected.[1] By Steinitz's theorem, as a 3-connected planar graph, it can be represented as the set of vertices and edges of a convex polyhedron; that is, it is a polyhedral graph. And, as with every polyhedral graph, its planar embedding is unique up to the choice of which of its faces is to be the outer face.[1]

Every Halin graph is a Hamiltonian graph, and every edge of the graph belongs to a Hamiltonian cycle. Moreover, any Halin graph remains Hamiltonian after deletion of any vertex.[4] Because every tree without vertices of degree 2 contains two leaves that share the same parent, every Halin graph contains a triangle. In particular, it is not possible for a Halin graph to be a triangle-free graph nor a bipartite graph.

More strongly, every Halin graph is almost pancyclic, in the sense that it has cycles of all lengths from 3 to n with the possible exception of a single even length. Moreover, any Halin graph remains almost pancyclic if a single edge is contracted, and every Halin graph without interior vertices of degree three is pancyclic.[5]

Every Halin graph has treewidth = 3.[6] Therefore, many graph optimization problems that are NP-complete for arbitrary planar graphs, such as finding a maximum independent set, may be solved in linear time on Halin graphs using dynamic programming.[7]

The weak dual of an embedded planar graph has vertices corresponding to bounded faces of the planar graph, and edges corresponding to adjacent faces. The weak dual of a Halin graph is always biconnected and outerplanar. This property may be used to characterize the Halin graphs: an embedded planar graph is a Halin graph, with the leaf cycle of the Halin graph as the outer face of the embedding, if and only if its weak dual is biconnected and outerplanar.[8]

The incidence chromatic number of a Halin graph G with maximum degree Δ(G) greater than four is Δ(G) + 1.[9] When Δ(G) = 3 or 4, the incidence chromatic number may be as large as 5 or 6 respectively.[10]

Computational complexity

It is possible to test whether a given n-vertex graph is a Halin graph in linear time, by finding a planar embedding of the graph (if one exists), and then testing whether there exists a face that has at least n/2 + 1 vertices, all of degree three. If so, there can be at most four such faces, and it is possible to check in linear time for each of them whether the rest of the graph forms a tree with the vertices of this face as its leaves. On the other hand, if no such face exists, then the graph is not Halin.[11] Alternatively, a graph with n vertices and m edges is Halin if and only if it is planar, 3-connected, and has a face whose number of vertices equals the circuit rank m − n + 1 of the graph, all of which can be checked in linear time.[8]

However, it is NP-complete to find the largest Halin subgraph of a given graph, to test whether there exists a Halin subgraph that includes all vertices of a given graph, or to test whether a given graph is a subgraph of a larger Halin graph.[12]

History

In 1971, Halin introduced the Halin graphs as a class of minimally 3-vertex-connected graphs: for every edge in the graph, the removal of that edge reduces the connectivity of the graph.[2] These graphs gained in significance with the discovery that many algorithmic problems that were computationally infeasible for arbitrary planar graphs could be solved efficiently on them.[4][8] This fact was later explained to be a consequence of their low treewidth, and of algorithmic meta-theorems like Courcelle's theorem that provide efficient solutions to these problems on any graph of low treewidth.[6][7]

Prior to Halin's work on these graphs, graph enumeration problems concerning the cubic Halin graphs were studied in 1856 by Thomas Kirkman[3] and in 1965 by Hans Rademacher.[13] Rademacher calls these graphs based polyhedra. He defines them as the cubic polyhedral graphs with f faces in which one of the faces has f − 1 sides. The graphs that fit this definition are exactly the cubic Halin graphs.

The Halin graphs are sometimes also called roofless polyhedra,[4] but, like "based polyhedra", this name may also refer to the cubic Halin graphs.[14] The convex polyhedra whose graphs are Halin graphs have also been called domes.[15]

References

  1. 1 2 3 Encyclopaedia of Mathematics, first Supplementary volume, 1988, ISBN 0-7923-4709-9, p. 281, article "Halin Graph", and references therein.
  2. 1 2 Halin, R. (1971), "Studies on minimally n-connected graphs", Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), London: Academic Press, pp. 129–136, MR 0278980.
  3. 1 2 Kirkman, Th. P. (1856), "On the enumeration of x-edra having triedral summits and an (x − 1)-gonal base", Philosophical Transactions of the Royal Society of London: 399–411, doi:10.1098/rstl.1856.0018, JSTOR 108592.
  4. 1 2 3 Cornuéjols, G.; Naddef, D.; Pulleyblank, W. R. (1983), "Halin graphs and the travelling salesman problem", Mathematical Programming 26 (3): 287–294, doi:10.1007/BF02591867.
  5. ↑ Skowrońska, Mirosława (1985), "The pancyclicity of Halin graphs and their exterior contractions", in Alspach, Brian R.; Godsil, Christopher D., Cycles in Graphs, Annals of Discrete Mathematics 27, Elsevier Science Publishers B.V., pp. 179–194.
  6. 1 2 Bodlaender, Hans (1988), Planar graphs with bounded treewidth (PDF), Technical Report RUU-CS-88-14, Department of Computer Science, Utrecht University.
  7. 1 2 Bodlaender, Hans (1988), "Dynamic programming on graphs with bounded treewidth", Proceedings of the 15th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 317, Springer-Verlag, pp. 105–118, doi:10.1007/3-540-19488-6_110, ISBN 978-3540194880.
  8. 1 2 3 Sysło, Maciej M.; Proskurowski, Andrzej (1983), "On Halin graphs", Graph Theory: Proceedings of a Conference held in Lagów, Poland, February 10–13, 1981, Lecture Notes in Mathematics 1018, Springer-Verlag, pp. 248–256, doi:10.1007/BFb0071635.
  9. ↑ Wang, Shu-Dong; Chen, Dong-Ling; Pang, Shan-Chen (2002), "The incidence coloring number of Halin graphs and outerplanar graphs", Discrete Mathematics 256 (1-2): 397–405, doi:10.1016/S0012-365X(01)00302-8, MR 1927561.
  10. ↑ Shiu, W. C.; Sun, P. K. (2008), "Invalid proofs on incidence coloring", Discrete Mathematics 308 (24): 6575–6580, doi:10.1016/j.disc.2007.11.030, MR 2466963.
  11. ↑ Fomin, Fedor V.; Thilikos, Dimitrios M. (2006), "A 3-approximation for the pathwidth of Halin graphs", Journal of Discrete Algorithms 4 (4): 499–510, doi:10.1016/j.jda.2005.06.004, MR 2577677.
  12. ↑ Horton, S. B.; Parker, R. Gary (1995), "On Halin subgraphs and supergraphs", Discrete Applied Mathematics 56 (1): 19–35, doi:10.1016/0166-218X(93)E0131-H, MR 1311302.
  13. ↑ Rademacher, Hans (1965), "On the number of certain types of polyhedra", Illinois Journal of Mathematics 9: 361–380, MR 0179682.
  14. ↑ Lovász, L.; Plummer, M. D. (1974), "On a family of planar bicritical graphs", Combinatorics (Proc. British Combinatorial Conf., Univ. Coll. Wales, Aberystwyth, 1973), London: Cambridge Univ. Press, pp. 103–107. London Math. Soc. Lecture Note Ser., No. 13, MR 0351915.
  15. ↑ Demaine, Erik D.; Demaine, Martin L.; Uehara, Ryuhei (2013), "Zipper unfolding of domes and prismoids", Proceedings of the 25th Canadian Conference on Computational Geometry (CCCG 2013), Waterloo, Ontario, Canada, August 8–10, 2013, pp. 43–48.

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