No-Bend Orthogonal Drawings of Subdivisions of Planar Triconnected Cubic Graphs

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1 N-B Oh Dw f Sv f P Tcc Cc Gh (Ex Ac) M. S Rh, N E, T Nhz G Sch f If Scc, Th Uvy, A-y 05, S , J. {,}@hz.c.h.c. h@c.h.c. Ac. A h h wh fx. I - h w f h, ch vx w ch w hz vc. A h hv - h w f f h - h w. I h w c c f h, c v f cc cc h, v - h x whh ch h G h - h w f f G h. 1 Ic A h w f h G w f G wh h v ch h ch vx, ch w c f hz vc, y w c xc h c [RN02,RNN02,RNN99,S84,T87]. A wh ch c w. If G h vx f fv, h G h h w. O h h h, f G h vx f fv, h, h x f G f, h G h h w, y. S h hv h w wh, whch ch w y hz vc. W c ch w - w. F 1() c - w f h h F. 1(). Rh. [RNN02] cy ffc c f h G wh 3 hv - w, v - h f - w f G h. W y h h G h - w f f h f G h - w. F 1(), (c) () c h f h ff f h h G. A h y h F. 1() h - w F. 1(). Th h h G h - w. I NP-c x whh h G wh 4 h - w G. L (E.): GD 2003, LNCS 2912, , c S-V B H 2004

2 388 M.S. Rh, N. E, T. Nhz c f h c f h f c () () (c) h h c f () () F. 1. A - w (), h ff (), (c) () f h h whch v f h cc cc h (). [GT01]. Hwv, f h G wh 3, D B. [DLV98] v O( 5 ) h f h w f G wh h f. Hc, y h h, c x O( 5 ) whh h wh 3 h - w. I h w c c f h, c v f cc cc h, v - h x whh ch h G h - w f - w f G h. Th h F. 1() f h cc cc h F. 1() y vc f w, hc v f h cc h. Th f h z fw. Sc 2 c f y. Sc 3 cy ffc c f v G f cc cc h hv - w; h c - h. Fy Sc 4 cc. 2 P I h c w v f y. L G =(V,E) cc h wh vx V E. Th (v) f vx v h f h f v G. W c vx f w G 2-vx f G. A h G c cc f (v) = 3 f vy vx v. FV V, G V h f

3 N-B Oh Dw f Sv 389 G y vc V h wh c h. F h G f G, w y G G h h f G y vc G. Th ccvy κ(g) f h G h f vc wh v cc h -vx h K 1. W y h G -cc f κ(g). Sv (, v) f h G h f h (, v) h (= w 0 ),w 1,w 2,,w, v(= w +1 ) hh w 2-vc w 1,w 2,,w, 1. A h G v f h G f G f G y v f h f G.A v f cc cc h cc, h f y vx h 2 3. L P = w 0,w 1,w 2,,w +1, 1, h f h G ch h (w 0 ) 3,(w 1 )=(w 2 )= = (w )=2, (w +1 ) 3. Th w c h h P = w 1,w 2,,w f P ch f G, w c vc w 0 w +1 h f h ch P.IfG v f cc h, h y 2-vx f G c xcy f h ch f G. L G cc h, Γ f G. Th c f fc f Γ cyc f G, y c fc fc cyc f Γ. W y F (Γ ) h fc f Γ. F cyc C f Γ, w c h h f Γ C (c C) h h Γ I (C) f C, c h h f Γ C (c C) h h Γ O (C) f C. Ay fc f Γ h Γ I (C) Γ O (C). A f G c f C f c xcy vx f C c C. A cyc C f Γ c - cyc f C h xcy Γ h whch w vc C c C. A f G c h f C f c xcy vx f C c C. A cyc C f Γ c -h cyc f C h xcy h Γ h whch w vc C c C. Bh h f f - cyc h f h f -h cyc G c c f G. Rh. [RNN02] cy ffc c f h G wh 3 hv - w, v h f - w f G f G h, fw. L 1. [RNN02] A Γ f cc h G wh 3 h - w f y f Γ f h fw h c: () h f 2-vc h fc F (Γ ); () vy 2- cyc Γ c w 2-vc; (c) vy 3- cyc Γ c 2-vx. Fh c x whh Γ f h c v, c f - w f Γ f Γ h. Ahh h v f h w, ffc x whh h h - w, c h

4 390 M.S. Rh, N. E, T. Nhz y hv x f. Hwv, h fw fc w f v f cc h. Fc 2 [NC88] L G v f cc h. Th h xcy f G f ch fc h fc. Fh, f y w Γ Γ f G, y fc cyc f Γ fc cyc f Γ. Fc 2 h v G f cc h h O() f, f ch ch fc. Ex y h h [RNN02] whh h h c L 1 h f ch f h O(), c x O( 2 ) whh h h G h - w. I Sc 3 w cy ffc c f G hv - w. O c - h x whh G f h c f - w f G h. W hv h fw fc v f cc cc h, whch w f Sc 3. L 3. [RNG02] L G v f cc cc h, Γ y f h h G. Th h fw () () h. () F y 2- cyc C f Γ, h f vc Γ I (C) c ch f G F (Γ ). () F y ch P F (Γ ), h fc f h h Γ P 2- cyc Γ. A cyc f Γ v C () (c) L 1 c cyc f Γ : 2- cyc f c 2-vx; 3- cyc f c 2-vx. L C 3-h cyc f Γ f G, F fc Γ I (C), Γ f G whch F F (Γ ). Th C 3- cyc f Γ.IfC c 2-vx, h C cyc f Γ. W h c 3-h cyc C f Γ 3-h cyc f Γ f c 2-vx, hw w c C 3-h cyc. 3 N-B Dw f P Gh W h fw cy ffc c f v G f cc cc h hv - w. Th 1. L G v f cc cc h, Γ y f G. Th h h G h - w f y f Γ h fc F fy h fw c () (v): () h f 2-vc F ;

5 N-B Oh Dw f Sv 391 () F c Γ I (C) f y 3- cyc C Γ ; () F c Γ O (C) f y 3-h cyc C Γ ; (v) f h xcy ch P F, h h fc F whch c P ff f F c w 2-vc whch P ; (v) f h xcy w ch P 1 P 2 F, f h, y P 1, c xcy vx, h h fc F whch c P 2 ff f F c 2-vx whch P 2. Pf. Ncy: A h G v f cc cc h h - w. Th G h Γ whch h - w. L F h fc f Γ ch h F = F (Γ ). Th w c hw h F f () (v), fw. () Sc Γ h - w, y L 1() h f 2-vc F (Γ )=F. () S f cc h F c Γ I (C) f 3- cyc C Γ. Th F c Γ O (C). O c v h C 3- cyc f Γ, hc y L 1 Γ hv - w, cc. () S f cc h F c Γ O (C) f 3-h cyc C Γ. Th F c Γ I (C). O c v h C 3- cyc f Γ, hc y L 1 Γ hv - w, cc. (v) A h h xcy ch P F. By L 3() Γ h 2- cyc C = F (Γ P ). Γ hv y 2- cyc h h C; hw, y L 3() F (Γ ) h ch h h P, hc F = F (Γ ) w hv w ch. Th Γ h xcy 2- cyc C. Sc Γ h - w, y L 1() C c w 2-vc. Sc F h xcy ch P, 2-vc F (Γ ) P hc h x h Q 1 f C h F (Γ ) c 2-vx. Th 2-vc C c h x h Q 2 f C h F (Γ ). Cy Q 2 F P. Hc F c w 2-vc whch P. (v) A h F h xcy w ch P 1 P 2 P 1 c xcy vx. Sc F (Γ )=F c f 2-vc y L 1(), P 2 c h 2-vc. By L 3() Γ h w 2- cyc C 1 = F (Γ P 1 ) C 2 = F (Γ P 2 ). By L 3() Γ hv y 2- cyc h h C 1 C 2. By L 1() C 2 c w 2-vc. Th x h Q 1 f C 2 h F (Γ ) c xcy 2-vx, hc h x h Q 2 f C 2 h F (Γ ) c 2-vx. Cy Q 2 F P 2, hc F c 2-vx whch P 2. Sffccy: A h Γ h fc F fy C () (v). L Γ f G ch h F = F (Γ ). W c v h h Γ f C () (c) L 1 hc Γ h - h w. Th f h f h x c. Q.E.D.

6 392 M.S. Rh, N. E, T. Nhz Th 1 y h x whh v G f cc cc h h - w f - w f G f x, fw. L Γ y f G. U h [RNN00,RNG02], c x whh Γ h fc F fy h c Th 1. If Γ h ch fc F, Γ ch h F = F (Γ ) c [NC88]. F - w f Γ [RNN02]. 4 Cc I h w cy ffc c f v G f cc cc h hv - w, v - h x whh G f h c f - w f G h. I f f w f - h f - w f c f h. Rfc [DLV98] G. D B, G. L F. V, Sy h w, SIAM J. C., 27(6), , [GT01] A. G R. T, O h c cxy f w c y, SIAM J. C., 31(2), , [NC88] T. Nhz N. Ch, P Gh: Thy Ah, Nh- H, A, [RN02] M. S. Rh T. Nhz, B- h w f 3-h, Pc. f WG 02, Lc. N C Scc, 2573, , [RNG02] M. S. Rh, T. Nhz, S. Ghh, Rc w f h, Pc. f GD 02, Lc, N C Scc, 2528, , 2002, J f Ah,. [RNN00] M. S. Rh, S. N T. Nhz, Bx-c w f h, J f Ah, 37, , [RNN02] M. S. Rh, M. Nz T. Nhz, Oh w f h wh, Pc. f Gh Dw 01, Lc. N C Scc, S-v, 2265, , 2002, J f Gh A. A.,. [RNN99] M.S. Rh, S. N T. Nhz, A h f h w f cc cc h, J f Gh A. A., h:// 3(4), , [S84] J. S, O -c, Nw, 14, , [T87] R. T, O h h wh h f, SIAM J. C., 16, , 1987.