S. MacLane. A combinatorial condition for planar graphs. Fundamenta Mathematicae, 28:22--32, 1937. Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
MacLane's Planarity Criterion for Locally Finite Graphs - Bruhn, Stein (2003) (3 citations) (Correct)
....nitely many nite cycles (the nite 2 regular connected subgraphs) We denote this Z 2 vector space by C n (G) A set (or family) H of subgraphs of G is called simple, if every edge of G lies in (the edge set of) at most two elements of H. Then, MacLane s criterion states: Theorem 1 (MacLane [9]) A nite graph G is planar if and only if C n (G) has a simple generating set. Wagner [13] raised the question if MacLane s result could be extended so that it characterises also planar graphs which are in nite. Rather than modifying the planarity criterion, Thomassen [12] describes all in ....
S. MacLane. A combinatorial condition for planar graphs. Fund. Math., 28:22-32, 1937.
Planarity of the 2-level Cactus Model - Cornelsen, Dinitz, Wagner (Correct)
....is based on properties of the set of projection paths of auxiliary edges, that is the set of (shortest) paths in the 2 level cactus between the end nodes of auxiliary edges. To obtain planarity, we give a new sucient planarity criterion, generalizing a corollary to the criterion of MacLane [11]. The question of planarity is not only of graph theoretical interest, it is also useful for algorithmic purposes. But the main reason, why we study planarity of the 2 level cactus, is an application in graph drawing. We are interested in drawing all small cuts ( bottlenecks ) of a graph. ....
.... . The set ZG of all cycles and unions of edge disjoint cycles is a subspace of the vector space EG and is called the cycle space of G. A 2 basis of G is a basis of the cycle space of G, such that every edge occurs in at most two elements of this basis. Theorem 1 (Planarity Criterion of MacLane [11]) A graph is planar if and only if it has a 2 basis. Moreover, any 2 basis of a 2 connected graph consists of all but one facial cycle of some of its planar representations. A short proof of MacLane s planarity criterion can be found in [13] A basis of the cycle space can be constructed from a ....
S. MacLane. A combinatorial condition for planar graphs. Fundamenta Mathematicae, 28:22-32, 1937.
Planarity of the 2-level Cactus Model - Cornelsen, Dinitz, Wagner (2001) (Correct)
....is based on properties of the set of projection paths of auxiliary edges, that is the set of (shortest) paths in the 2 level cactus between the end nodes of auxiliary edges. To obtain planarity, we give a new sucient planarity criterion, generalizing a corollary to the criterion of MacLane [10]. The question of planarity is not only of graph theoretical interest, it is also useful for algorithmic purposes. But the main reason, why we study planarity of the 2 level cactus, is 1 an application in graph drawing. We are interested in drawing all small cuts ( bottlenecks ) of a graph. ....
.... . The set ZG of all cycles and unions of edge disjoint cycles is a subspace of the vector space EG and is called the cycle space of G. A 2 basis of G is a basis of the cycle space of G, such that every edge occurs in at most two elements of this basis. Theorem 1 (Planarity Criterion of MacLane [10]) A graph is planar if and only if it has a 2 basis. Moreover, any 2 basis of a 2 connected graph consists of all but one facial cycle of some of its planar representations. A short proof of MacLane s planarity criterion can be found in [12] A basis of the cycle space can be constructed from a ....
S. MacLane. A combinatorial condition for planar graphs. Fundamenta Mathematicae, 28:22-32, 1937.
Planarizing Graphs - A Survey and Annotated Bibliography - Liebers (2001) (6 citations) (Correct)
....graph and the face of a polytope. A. Liebers, Planarizing Graphs , JGAA, 5(1) 1 74 (2001) 9 Theorem 6 (Wagner [Wag37a] Harary and Tutte [HT65] A graph G is planar if and only if it does not contain K 5 or K 3,3 as a minor. For further characterizations of planar graphs see for example [Whi33, Mac37] Sch89, dFdM96] NC88, BS93, Kel93, ABL95] dV90, dV93, Sch97] and [TT97] An algorithm for determining whether a given graph is planar was first developed by Auslander and Parter [AP61] and Goldstein [Gol63] Hopcroft and Tarjan [HT74] improved it to run in linear time. Wil80] and [Mut94, ....
S. MacLane. A combinatorial condition for planar graphs. Fundamenta Mathematicae, 28:22--32, 1937.
Optimizing over All Combinatorial Embeddings of a Planar Graph - Mutzel, Weiskircher (1998) (2 citations) (Correct)
....of feasible solutions corresponds to the set of all possible combinatorial embeddings of a given biconnected planar graph. One way of constructing such an integer linear program is by using the fact that every combinatorial embedding corresponds to a 2 fold complete set of circuits (see MacLane [11]) The variables in such a program are all simple cycles in the graph; the constraints guarantee that the chosen subset of all simple cycles is complete and that no edge of the graph appears in more than two simple cycles of the subset. We have chosen another way of formulating the problem. The ....
S. MacLane. A combinatorial condition for planar graphs. Fundamenta Mathematicae, 28:22--32, 1937.
Optimizing Over All Combinatorial Embeddings of a Planar.. - Mutzel, Weiskircher (1999) (2 citations) (Correct)
....set of feasible solutions corresponds to the set of all possible combinatorial embeddings of a given biconnected planar graph. One way of constructing such an integer linear program is by using the fact that every combinatorial embedding corresponds to a 2 fold complete set of cycles (see MacLane [11]) The variables in such a program are all simple cycles in the graph; the constraints guarantee that the chosen subset of all simple cycles is complete and that no edge of the graph appears in more than two simple cycles of the subset. We have chosen another way of formulating the problem. The ....
S. MacLane. A combinatorial condition for planar graphs. Fundamenta Mathematicae, 28:22--32, 1937.
A New Parallel Algorithm for Planarity Testing - Bader, Sreshta (2003) (Correct)
No context found.
S. MacLane. A combinatorial condition for planar graphs. Fundamenta Mathematicae, 28:22--32, 1937.
The Cycle Space of an Infinite Graph - Diestel (1 citation) (Correct)
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S. MacLane, A combinatorial condition for planar graphs, Fund. Math 28 (1937), 22--32.
Planar Graphs with Topological Constraints - Dornheim (2002) (1 citation) (Correct)
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S. MacLane. A combinatorial condition for planar graphs. Fundam. Math., 28: 22--32, 1937
The Cycle Space of an Infinite Graph - Diestel (2004) (1 citation) (Correct)
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S. MacLane, A combinatorial condition for planar graphs, Fund. Math 28 (1937), 22--32.
Optimizing over All Combinatorial Embeddings of a Planar.. - Mutzel, Weiskircher (1999) (2 citations) (Correct)
No context found.
S. MacLane. A combinatorial condition for planar graphs. Fundamenta Mathe- maticae, 28:22-32, 1937.
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