Supplement to Internally 4-connected projective-planar graphs
نویسنده
چکیده
A5 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {6, 7}, {6, 8}, {6, 9}, {6, 10}, {7, 8}, {7, 9}, {7, 10}, {8, 9}, {8, 10}, {9, 10}} C11 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {6, 9}, {6, 10}, {6, 11}, {7, 9}, {7, 10}, {7, 11}, {8, 9}, {8, 10}, {8, 11}} E42 = {{1, 4}, {1, 5}, {1, 6}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {7, 10}, {7, 11}, {7, 12}, {8, 10}, {8, 11}, {8, 12}, {9, 10}, {9, 11}, {9, 12}} A1 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {5, 6}, {5, 7}, {5, 8}, {5, 9}, {6, 7}, {6, 8}, {6, 9}, {7, 8}, {7, 9}, {8, 9}} C1 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {5, 6}, {5, 7}, {5, 8}, {6, 9}, {6, 10}, {7, 9}, {7, 10}, {8, 9}, {8, 10}} E1 = {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 6}, {6, 7}, {6, 8}, {6, 9}, {7, 10}, {7, 11}, {8, 10}, {8, 11}, {9, 10}, {9, 11}} B3 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {4, 7}, {4, 8}, {5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}} C2 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {4, 7}, {4, 8}, {5, 6}, {5, 7}, {5, 8}, {6, 9}, {7, 9}, {8, 9}} D1 = {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}, {5, 8}, {5, 9}, {6, 7}, {6, 8}, {6, 9}, {7, 10}, {8, 10}, {9, 10}} D4 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {4, 8}, {5, 7}, {5, 9}, {6, 7}, {6, 9}, {7, 8}, {8, 9}} E6 = {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}, {5, 9}, {6, 8}, {6, 10}, {7, 9}, {7, 11}, {10, 11}, {10, 13}, {11, 12}, {12, 13}} F6 = {{1, 2}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 4}, {3, 5}, {4, 6}, {5, 7}, {5, 9}, {6, 8}, {6, 10}, {7, 8}, {7, 10}, {8, 9}, {9, 10}} B1 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} C7 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {4, 8}, {5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}} D3 = {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {4, 7}, {4, 8}, {5, 6}, {5, 7}, {5, 8}, {6, 7}, {6, 8}, {7, 8}} D9 = {{1, 3}, {1, 4}, {1, 7}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 6}, {4, 5}, {4, 9}, {4, 10}, {5, 8}, {6, 7}, {6, 8}, {7, 9}, {7, 10}, {8, 9}, {8, 10}} D12 = {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 9}, {4, 6}, {4, 7}, {4, 8}, {5, 7}, {5, 9}, {6, 7}, {6, 8}, {7, 8}, {8, 9}} E3 = {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {3, 8}, {4, 6}, {4, 7}, {4, 8}, {5, 6}, {5, 7}, {5, 8}} E5 = {{1, 3}, {1, 4}, {1, 6}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 5}, {4, 8}, {4, 9}, {5, 6}, {5, 7}, {6, 8}, {6, 9}, {7, 8}, {7, 9}} E11 = {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 8}, {4, 9}, {5, 7}, {5, 8}, {6, 10}, {7, 9}, {8, 10}, {9, 10}} E19 = {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 7}, {5, 8}, {5, 9}, {6, 7}, {6, 8}, {6, 9}, {7, 8}, {7, 9}, {8, 9}} E27 = {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 7}, {3, 8}, {4, 6}, {4, 10}, {5, 8}, {5, 9}, {6, 7}, {6, 9}, {7, 10}, {8, 10}, {9, 10}} F1 = {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {4, 7}, {5, 8}, {5, 9}, {6, 8}, {6, 9}, {7, 8}, {7, 9}} G1 = {{1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 7}, {5, 8}, {6, 9}, {6, 10}, {7, 9}, {7, 10}, {8, 9}, {8, 10}}
منابع مشابه
Internally 4-connected projective-planar graphs
5 Archdeacon proved that projective-planar graphs are characterized by 35 excluded minors. 6 Using this result we show that internally 4-connected projective-planar graphs are characterized 7 by 23 internally 4-connected excluded minors. 8
متن کاملProjective-Planar Graphs with no K3, 4-Minor. II
The authors previously published an iterative process to generate a class of projectiveplanarK3,4-free graphs called ‘patch graphs’. They also showed that any simple, almost 4-connected, nonplanar, and projective-planar graph that is K3,4-free is a subgraph of a patch graph. In this paper, we describe a simpler and more natural class of cubic K3,4free projective-planar graphs which we call Möbi...
متن کاملHall-Type Results for 3-Connected Projective Graphs
4 Projective planar graphs can be characterized by a set A of 35 excluded minors. However, 5 these 35 are not equally important. A set E of 3-connected members of A is excludable if there 6 are only finitely many 3-connected non-projective planar graphs that do not contain any graph 7 in E as a minor. In this paper we show that there are precisely two minimal excludable sets, 8 which have sizes...
متن کامل$n$-Array Jacobson graphs
We generalize the notion of Jacobson graphs into $n$-array columns called $n$-array Jacobson graphs and determine their connectivities and diameters. Also, we will study forbidden structures of these graphs and determine when an $n$-array Jacobson graph is planar, outer planar, projective, perfect or domination perfect.
متن کاملProjective-Planar Graphs with No K3, 4-Minor
There are several graphs H for which the precise structure of graphs that do not contain a minor isomorphic to H is known. In particular, such structure theorems are known for K5 [13], V8 [8] and [7], the cube [3], the octahedron [4], and several others. Such characterizations can often be very useful, e.g., Hadwiger’s conjecture for k = 4 is verified by using the structure for K5-free graphs, ...
متن کاملExtremal Results in Sparse Pseudorandom Graphs Jacob
3-Connected Minor Minimal Non-Projective Planar Graphs with an Internal 3-Separation Arash Asadi, Georgia Institute of Technology The property that a graph has an embedding in the projective plane is closed under taking minors. So by the well known theorem of Robertson and Seymour, there exists a finite list of minor-minimal graphs, call it L, such that a given graph G is projective planar if a...
متن کامل