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The minimum spanning tree problem on a planar graph

Tomomi Matsui

1995
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Discrete Applied Mathematics
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Finding a spanning tree of minimum weight is one of the best known graph problems. Several efficient algorithms exist for solving this problem [3] [4] [5] 7, 9] . This note presents a linear time algorithm for the minimum spanning tree problem on a planar graph. Cheriton and Tarjan [l] have proposed a linear time algorithm for this problem. The time complexity of our algorithm is the same as that of Cheriton and Tarjan's algorithm. Different from Cheriton and Tarjan's algorithm, our algorithm
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... es not require the clean-up activity. So, the implementation of our algorithm is very easy. Our algorithm maintains a pair of a planar graph and its dual graph and breeds both a minimum spanning tree of the original graph and a maximum spanning tree of a dual graph. In each iteration of our algorithm, either the number of edges decreases or a vertex of the planar graph or its dual graph is deleted. By employing a simple bucket structure, we can save the time complexity of every iteration to O(1). Let us consider an undirected graph G = (I/, E) with the vertex set V and the edge set E. For any vertex u of G, 6,(v) denotes the set of edges in G incident to v. For any edge subset E' c E, the graph (V, E') is called a spanning forest of G when the graph does not contain any cycle. A spanning forest of G is called a spanning tree when it is connected. In this note, we present a spanning forest as its edge set. Given a graph G and its edge e, G\e denotes the graph obtained by deleting the edge e and G/e denotes the graph obtained by contracting e. For each edge e E E, w(e) denotes the weight of the edge e. The weight of an edge subset F, denoted by w(F), is the sum of the weights of edges in F. A maximal spanning forest F is called a minimum (maximum) weight spanningforest, when F minimizes (maximizes) the weight w(F). A graph is called planar if it can be drawn in the plane so that its edges intersect only at their ends. Given a graph G = (V, E) , a graph G* = (V *, E) with common edge set 0166-218X/95/$09.50 0 1995-Elsevier Science B.V. All rights reserved SSDI 0166-218X(94)00095-6

doi:10.1016/0166-218x(94)00095-u
fatcat:rbmyp6ga3jga5m2k5qqnxby4mi