Caterpillar tree

This article is about graph theory. For the shrub, see Plumeria alba.
A caterpillar

In graph theory, a caterpillar or caterpillar tree is a tree in which all the vertices are within distance 1 of a central path.

Caterpillars were first studied in a series of papers by Harary and Schwenk. The name was suggested by A. Hobbs.[1][2] As Harary & Schwenk (1973) colorfully write, "A caterpillar is a tree which metamorphoses into a path when its cocoon of endpoints is removed."[1]

Equivalent characterizations

The following characterizations all describe the caterpillar trees:

Generalizations

A k-tree is a chordal graph with exactly n k maximal cliques, each containing k + 1 vertices; in a k-tree that is not itself a (k + 1)-clique, each maximal clique either separates the graph into two or more components, or it contains a single leaf vertex, a vertex that belongs to only a single maximal clique. A k-path is a k-tree with at most two leaves, and a k-caterpillar is a k-tree that can be partitioned into a k-path and some k-leaves, each adjacent to a separator k-clique of the k-path. In this terminology, a 1-caterpillar is the same thing as a caterpillar tree, and k-caterpillars are the edge-maximal graphs with pathwidth k.[6]

A lobster graph is a tree in which all the vertices are within distance 2 of a central path.[8]

Enumeration

Caterpillars provide one of the rare graph enumeration problems for which a precise formula can be given: when n  3, the number of caterpillars with n unlabeled vertices is [1]

2^{n-4}+2^{\lfloor (n-4)/2\rfloor}.

For n = 1, 2, 3, ... the numbers of n-vertex caterpillars are

1, 1, 1, 2, 3, 6, 10, 20, 36, 72, 136, 272, 528, 1056, 2080, 4160, ... (sequence A005418 in OEIS).

Computational complexity

Finding a spanning caterpillar in a graph is NP-complete. A related optimization problem is the Minimum Spanning Caterpillar Problem (MSCP) , where a graph has dual costs over its edges and the goal is to find a caterpillar tree that spans the input graph and has the smallest overall cost. Here the cost of the caterpillar is defined as the sum of the costs of its edges, where each edge takes one of the two costs based on its role as a leaf edge or an internal one. There is no f(n)-approximation algorithm for the MSCP unless P = NP. Here f(n) is any polynomial-time computable function of n, the number of nodes of a graph.[9]

There is a parametrized algorithm that finds an optimal solution for the MSCP in bounded treewidth graphs. So both the Spanning Caterpillar Problem and the MSCP have linear time algorithms if a graph is an outerplanar, a series-parallel, or a Halin graph.[9]

Applications

Caterpillar trees have been used in chemical graph theory to represent the structure of benzenoid hydrocarbon molecules. In this representation, one forms a caterpillar in which each edge corresponds to a 6-carbon ring in the molecular structure, and two edges are incident at a vertex whenever the corresponding rings belong to a sequence of rings connected end-to-end in the structure. El-Basil (1987) writes, "It is amazing that nearly all graphs that played an important role in what is now called "chemical graph theory" may be related to caterpillar trees." In this context, caterpillar trees are also known as benzenoid trees and Gutman trees, after the work of Ivan Gutman in this area.[2][10][11]

References

  1. 1.0 1.1 1.2 Harary, Frank; Schwenk, Allen J. (1973), "The number of caterpillars", Discrete Mathematics 6 (4): 359–365, doi:10.1016/0012-365x(73)90067-8.
  2. 2.0 2.1 2.2 El-Basil, Sherif (1987), "Applications of caterpillar trees in chemistry and physics", Journal of Mathematical Chemistry 1 (2): 153–174, doi:10.1007/BF01205666.
  3. 3.0 3.1 3.2 3.3 Harary, Frank; Schwenk, Allen J. (1971), "Trees with Hamiltonian square", Mathematika 18: 138–140, doi:10.1112/S0025579300008494.
  4. Harary, Frank; Schwenk, Allen J. (1972), "A new crossing number for bipartite graphs", Utilitas Math. 1: 203–209.
  5. Raychaudhuri, Arundhati (1995), "The total interval number of a tree and the Hamiltonian completion number of its line graph", Information Processing Letters 56 (6): 299–306, doi:10.1016/0020-0190(95)00163-8.
  6. 6.0 6.1 Proskurowski, Andrzej; Telle, Jan Arne (1999), "Classes of graphs with restricted interval models", Discrete Mathematics and Theoretical Computer Science 3: 167–176.
  7. Eckhoff, Jürgen (1993), "Extremal interval graphs", Journal of Graph Theory 17 (1): 117–127, doi:10.1002/jgt.3190170112.
  8. Weisstein, Eric W., "Lobster", MathWorld.
  9. 9.0 9.1 Khosravani, Masoud (2011). Searching for optimal caterpillars in general and bounded treewidth graphs (Ph.D.). University of Auckland.
  10. Gutman, Ivan (1977), "Topological properties of benzenoid systems", Theoretica Chimica Acta 45 (4): 309–315, doi:10.1007/BF00554539.
  11. El-Basil, Sherif (1990), "Caterpillar (Gutman) trees in chemical graph theory", in Gutman, I.; Cyvin, S. J., Advances in the Theory of Benzenoid Hydrocarbons, Topics in Current Chemistry 153, pp. 273–289, doi:10.1007/3-540-51505-4_28.

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