Graph power

Quality 0.50 · 3 views · Updated 2 months ago

Graph of short distances in another graph

title: "Graph power" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["graph-operations"] description: "Graph of short distances in another graph" topic_path: "general/graph-operations" source: "https://en.wikipedia.org/wiki/Graph_power" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Graph of short distances in another graph ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/bd/Square_of_a_graph.svg" caption="The square of a graph"] ::

In graph theory, a branch of mathematics, the kth power G of an undirected graph G is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in G is at most k. Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: G is called the square of G, G is called the cube of G, etc.

Graph powers should be distinguished from the products of a graph with itself, which (unlike powers) generally have many more vertices than the original graph.

Properties

If a graph has diameter d, then its d-th power is the complete graph. If a graph family has bounded clique-width, then so do its d-th powers for any fixed d.{{citation | last = Todinca | first = Ioan | contribution = Coloring powers of graphs of bounded clique-width | doi = 10.1007/978-3-540-39890-5_32 | mr = 2080095 | pages = 370–382 | publisher = Springer, Berlin | series = Lecture Notes in Comput. Sci. | title = Graph-theoretic concepts in computer science | volume = 2880 | year = 2003 | isbn = 978-3-540-20452-7

Coloring

Graph coloring on the square of a graph may be used to assign frequencies to the participants of wireless communication networks so that no two participants interfere with each other at any of their common neighbors, and to find graph drawings with high angular resolution.{{citation | last1 = Formann | first1 = M. | last2 = Hagerup | first2 = T. | last3 = Haralambides | first3 = J. | last4 = Kaufmann | first4 = M. | last5 = Leighton | first5 = F. T. | author5-link = F. Thomson Leighton | last6 = Symvonis | first6 = A. | last7 = Welzl | first7 = E. | author7-link = Emo Welzl | last8 = Woeginger | first8 = G. | author8-link = Gerhard J. Woeginger | doi = 10.1137/0222063 | issue = 5 | journal = SIAM Journal on Computing | mr = 1237161 | pages = 1035–1052 | title = Drawing graphs in the plane with high resolution | volume = 22 | year = 1993}}.

Both the chromatic number and the degeneracy of the kth power of a planar graph of maximum degree Δ are O(Δ), where the degeneracy bound shows that a greedy coloring algorithm may be used to color the graph with this many colors.{{citation | last1 = Agnarsson | first1 = Geir | last2 = Halldórsson | first2 = Magnús M. | contribution = Coloring powers of planar graphs | location = San Francisco, California, USA | pages = 654–662 | title = Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '00) | year = 2000}}. For the special case of a square of a planar graph, Wegner conjectured in 1977 that the chromatic number of the square of a planar graph is at most max(Δ + 5, + 1), and it is known that the chromatic number is at most + O(1).{{citation | last1 = Kramer | first1 = Florica | last2 = Kramer | first2 = Horst | doi = 10.1016/j.disc.2006.11.059 | issue = 2–3 | journal = Discrete Mathematics | mr = 2378044 | pages = 422–426 | title = A survey on the distance-colouring of graphs | volume = 308 | year = 2008| doi-access = free | last1 = Molloy | first1 = Michael | last2 = Salavatipour | first2 = Mohammad R. | doi = 10.1016/j.jctb.2004.12.005 | issue = 2 | journal = Journal of Combinatorial Theory | series = Series B | mr = 2145512 | pages = 189–213 | title = A bound on the chromatic number of the square of a planar graph | volume = 94 | year = 2005| hdl = 1807/9473 | hdl-access = free

Although the chromatic number of the square of a nonplanar graph with maximum degree Δ may be proportional to Δ2 in the worst case, it is smaller for graphs of high girth, being bounded by O(Δ2 / log Δ) in this case.{{citation | last1 = Alon | first1 = Noga | author1-link = Noga Alon | last2 = Mohar | first2 = Bojan | author2-link = Bojan Mohar | doi = 10.1017/S0963548301004965 | issue = 1 | journal = Combinatorics, Probability and Computing | mr = 1888178 | pages = 1–10 | title = The chromatic number of graph powers | volume = 11 | year = 2002| s2cid = 2706926 }}.

Determining the minimum number of colors needed to color the square of a graph is NP-hard, even in the planar case.

Hamiltonicity

The cube of every connected graph necessarily contains a Hamiltonian cycle. It is not necessarily the case that the square of a connected graph is Hamiltonian, and it is NP-complete to determine whether the square is Hamiltonian.{{citation | last = Underground | first = Polly | authorlink = Václav Chvátal | doi = 10.1016/0012-365X(78)90164-4 | issue = 3 | journal = Discrete Mathematics | mr = 522906 | page = 323 | title = On graphs with Hamiltonian squares | volume = 21 | year = 1978| doi-access = free | last = Diestel | first = Reinhard | contribution = 10. Hamiltonian cycles | edition = corrected 4th electronic | title = Graph Theory | url = https://www.math.uni-hamburg.de/home/diestel/books/graph.theory/preview/Ch10.pdf | year = 2012}}.

Computational complexity

The kth power of a graph with n vertices and m edges may be computed in time O(mn) by performing a breadth first search starting from each vertex to determine the distances to all other vertices, or slightly faster using more sophisticated algorithms.{{citation | last = Chan | first = Timothy M. | doi = 10.1145/2344422.2344424 | issue = 4 | journal = ACM Transactions on Algorithms | mr = 2981912 | pages = A34:1–A34:17 | title = All-pairs shortest paths for unweighted undirected graphs in o(mn) time | volume = 8 | year = 2012| s2cid = 1212001

Given a graph, deciding whether it is the square of another graph is NP-complete. | last1 = Motwani | first1 = R. | last2 = Sudan | first2 = M. | journal = Discrete Applied Mathematics | pages = 81–88 | title = Computing roots of graphs is hard | volume = 54 | year = 1994 | doi=10.1016/0166-218x(94)00023-9| doi-access = free Moreover, it is NP-complete to determine whether a graph is a kth power of another graph, for a given number k ≥ 2, or whether it is a kth power of a bipartite graph, for k 2.{{citation | last1 = Le | first1 = Van Bang | last2 = Nguyen | first2 = Ngoc Tuy | contribution = Hardness results and efficient algorithms for graph powers | doi = 10.1007/978-3-642-11409-0_21 | location = Berlin | mr = 2587715 | pages = 238–249 | publisher = Springer | series = Lecture Notes in Computer Science | title = Graph-Theoretic Concepts in Computer Science: 35th International Workshop, WG 2009, Montpellier, France, June 24-26, 2009, Revised Papers | volume = 5911 | year = 2010 | isbn = 978-3-642-11408-3}}.

In directed graphs

::figure[src="https://upload.wikimedia.org/wikipedia/commons/1/19/Square_of_digraph.svg" caption="digraph"] ::

For the square of directed graphs (or digraphs), each pair of vertices connected by a directed path of length two become connected by a path in the same direction of length one. Pairs of points directed to the same vertex, then, do not cause a connection between them in the square of a directed graph.

The second neighborhood problem can be stated in terms of the square of a digraph, asking if there exists a vertex in every oriented graph whose degree increase by at least a factor of two when the graph is squared.{{citation | last1 = Dean | first1 = Nathaniel | last2 = Latka | first2 = Brenda J. | department = Proceedings of the Twenty-Sixth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1995) | journal = Congressus Numerantium | mr = 1369296 | pages = 73–80 | title = Squaring the tournament—an open problem | volume = 109 | year = 1995}}

Induced subgraphs

::figure[src="https://upload.wikimedia.org/wikipedia/commons/9/93/Demi-3-cube.svg" caption="''K''₄}} as the half-square of a [[cube graph"] ::

The half-square of a bipartite graph G is the subgraph of G2 induced by one side of the bipartition of G. Map graphs are the half-squares of planar graphs,{{citation | last1 = Chen | first1 = Zhi-Zhong | last2 = Grigni | first2 = Michelangelo | last3 = Papadimitriou | first3 = Christos H. | author3-link = Christos Papadimitriou | doi = 10.1145/506147.506148 | issue = 2 | journal = Journal of the ACM | mr = 2147819 | pages = 127–138 | title = Map graphs | volume = 49 | year = 2002| arxiv = cs/9910013| s2cid = 2657838 | last = Shpectorov | first = S. V. | doi = 10.1006/eujc.1993.1016 | issue = 2 | journal = European Journal of Combinatorics | mr = 1206617 | pages = 117–130 | title = On scale embeddings of graphs into hypercubes | volume = 14 | year = 1993| doi-access = free

Leaf powers are the subgraphs of powers of trees induced by the leaves of the tree. A k-leaf power is a leaf power whose exponent is k.{{citation | last1 = Nishimura | first1 = N. | last2 = Ragde | first2 = P. | last3 = Thilikos | first3 = D.M. | journal = Journal of Algorithms | pages = 69–108 | title = On graph powers for leaf-labeled trees | volume = 42 | year = 2002 | doi=10.1006/jagm.2001.1195}}.

References

(2008). "Graph Theory". Springer.
"Graph Power".
{{harvtxt. Agnarsson. Halldórsson. 2000 list publications proving NP-hardness for general graphs by McCormick (1983) and Lin and Skiena (1995), and for planar graphs by Ramanathan and Lloyd (1992, 1993).
{{harvtxt. Bondy. Murty. 2008, p. 105.
(2011). "Handbook of Product Graphs". CRC Press.

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::