By Ioannis G Tollis, Giuseppe Liotta PH., Roberto Tamassia
This e-book comprises quantity 7 of the "Journal of Graph Algorithms and functions" (JGAA). JGAA is a peer-reviewed medical magazine dedicated to the booklet of high quality study papers at the research, layout, implementation, and purposes of graph algorithms. components of curiosity comprise computational biology, computational geometry, special effects, computer-aided layout, laptop and interconnection networks, constraint platforms, databases, graph drawing, graph embedding and structure, wisdom illustration, multimedia, software program engineering, telecommunications networks, consumer interfaces and visualization, and VLSI circuit layout. "Graph Algorithms and purposes four" offers contributions from widespread authors and comprises chosen papers from the 7th foreign Workshop on Algorithms and information constructions (WADS 2001) and the 2001 Symposium on Graph Drawing (GD 2001). All papers within the publication have broad diagrams and provide a different remedy of graph algorithms targeting the real functions.
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7) 5 (Cr × C3 ) K6 \ M2 4 (Cr × C3 ) K4 3 (Thm. 2) 4 (Cr × C3 ) K5 \ M2 4 (Thm. 5) 3r 3r 3r 3r 3r 3r (3 · 15 + 6)r (3 · 19 + 6)r (3 · 10 + 6)r (3 · 13 + 6)r (3 · 6 + 6)r (3 · 8 + 6)r 4-Connected Multigraphs 4 (Lem. 7) 5 Cr × ( 32 · C4 ) 6 (Lem. 7) 6 Cr × (2 · C3 ) r r 10r 9r 4-Connected Pseudographs 3 6 (Cr × C3 ) L1 3r (3 · 1 + 6)r continued on next page 2 3 2 2·12 8 33 = 11 2·14 28 41 = 41 2·7 14 23 = 23 2·8 16 29 = 29 2·3 2 15 = 5 2·4 8 19 = 19 2 9 =1 = 1 2 3·12 12 51 = 17 3·14 2 63 = 3 3·7 7 36 = 12 3·8 8 45 = 15 3·3 3 24 = 8 3·4 2 30 = 5 4 2 10 = 5 6 2 9 = 3 9 9 =1 D.
We now provide a formal prove of the well-known result that the maximum number of bends per edge in the drawing in Figure 22(a) is optimal. Theorem 10. Every drawing of 6 · K2 has an edge with at least three bends. Proof. Let the vertices of 6 · K2 be v and w. Since 6 · K2 is 6-regular every port at v and w is used. The two vertices are either (a) collinear, (b) coplanar but not collinear, or (c) not coplanar, as illustrated in Figure 23. D. Wood, Lower Bounds for 3-D Drawings, JGAA, 7(1) 33–77 (2003) ´ µ 54 ´ µ Figure 22: Drawings of 6 · K2 with (a) a maximum of three bends per edge, and (b) a total of twelve bends.
Figure 10: 7-sided 0-bend drawings of C7 . Proof. Consider an arbitrary, but fixed, 7-sided 0-bend drawing of C7 . By Observation 2, there is no 2-dimensional 0-bend drawing of an odd cycle, and if there is an I-side in a drawing of a cycle for some I ∈ {X, Y, Z}, then there are at least two I-sides. Therefore in a 7-sided cycle, without loss of generality three of the sides are X-sides, two are Y -sides and two are Z-sides. Clearly the length of one of the X-sides equals the sum of the lengths of the other two X-sides.