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**1 - 4**of**4**### Enumerating Rooted Graphs with . . .

, 2009

"... In this paper, we consider an arbitrary class H of rooted graphs such that each biconnected component is given by a representation with reflectional symmetry, which allows a rooted graph to have several different representations, called embeddings. We give a general framework to design algorithms fo ..."

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In this paper, we consider an arbitrary class H of rooted graphs such that each biconnected component is given by a representation with reflectional symmetry, which allows a rooted graph to have several different representations, called embeddings. We give a general framework to design algorithms for enumerating embeddings of all graphs in H without repetition. The framework yields an efficient enumeration algorithm for a class H if the class B of biconnected graphs used in the graphs in H admits an efficient enumeration algorithm. For example, for the class B of rooted cycles, we can easily design an algorithm of enumerating rooted cycles so that delivers the difference between two consecutive cycles in constant time in a series of all outputs. Hence our framework implies that, for the class H of all rooted cacti, there is an algorithm that enumerates each cactus in constant time.

### Enumerating biconnected rooted plane graphs

, 2010

"... A plane graph is a drawing of a planar graph in the plane such that no two edges cross each other. A rooted plane graph has a designated outer vertex. For given positive integers n and g, let G2(n, g) denote the set of all biconnected rooted plane graphs with exactly n vertices such that the size of ..."

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A plane graph is a drawing of a planar graph in the plane such that no two edges cross each other. A rooted plane graph has a designated outer vertex. For given positive integers n and g, let G2(n, g) denote the set of all biconnected rooted plane graphs with exactly n vertices such that the size of each inner face is at most g. In this paper, we give an algorithm that enumerates all plane graphs in G2(n, g). The algorithm runs in constant time per each by outputting the difference from the previous output.

### Listing Triconnected Rooted Plane Graphs

, 2010

"... A plane graph is a drawing of a planar graph in the plane such that no two edges cross each other. A rooted plane graph has a designated outer vertex. For given positive integers n ≥ 1 and g ≥ 3, let G3(n, g) denote the set of all triconnected rooted plane graphs with exactly n vertices such that th ..."

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A plane graph is a drawing of a planar graph in the plane such that no two edges cross each other. A rooted plane graph has a designated outer vertex. For given positive integers n ≥ 1 and g ≥ 3, let G3(n, g) denote the set of all triconnected rooted plane graphs with exactly n vertices such that the size of each inner face is at most g. In this paper, we give an algorithm that enumerates all plane graphs in G3(n, g). The algorithm runs in constant time per each by outputting the difference from the previous output.

### Generating internally triconnected . . .

, 2010

"... A biconnected plane graph G is called internally triconnected if any cut-pair consists of outer vertices and its removal results in only components each of which contains at least one outer vertex. In a rooted plane graph, an edge is designated as an outer edge with a specified direction. For given ..."

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A biconnected plane graph G is called internally triconnected if any cut-pair consists of outer vertices and its removal results in only components each of which contains at least one outer vertex. In a rooted plane graph, an edge is designated as an outer edge with a specified direction. For given positive integers n ≥ 1 and g ≥ 3, let G3(n, g) (resp., Gint(n, g)) denote the class of all triconnected (resp., internally triconnected) rooted plane graphs with exactly n vertices such that the size of each inner face is at most g. In this paper, we present an efficient algorithm that enumerates all rooted plane graphs in Gint(n, g) − G3(n, g) in O(n) space and in O(1)-time delay.