# 699-approximation algorithm for Max-Bisection by Ye Y.

By Ye Y.

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**Sample text**

Suppose that Condition 1 of Lemma 5 holds. Denote by Cu,v , by Cu,z , and by Cv,z the clustered graphs whose underlying graphs Gu,v , Gu,z , and Gv,z are the subgraphs of G induced by the vertices incident to and internal to cycles Cu,v ≡ (u, v) ∪ (Pu \ {u1 }) ∪ (u2 , v2 ) ∪ (Pv \ {v1 }), Cu,z ≡ (u, z) ∪ Pu ∪ Pz , and Cv,z ≡ (v, z) ∪ Pv ∪ Pz , and whose inclusion trees Tu,v , Tu,z , and Tv,z are the subtrees of T induced by the clusters containing vertices of Gu,v , Gu,z , and Gv,z . Straight-Line Rectangular Drawings of Clustered Graphs z2=zZ−1 z=zZ σ(u,v,z) z3 z z2 33 z=z Z u1=v1=z1 u1=v1=z1=zZ* Cu,z Cv,z v u u2 u3 v2 v3 v4 v=vV u=uU (a) u2 u3 u=uU u U−1 v2 C u,v v V−1 v=v V (b) Fig.

All clusters diﬀerent from the root do not contain smaller clusters, admit straight-line convex drawings and straight-line rectangular drawings in polynomial area. 36 P. Angelini, F. Frati, and M. Kaufmann References 1. : Straight-line rectangular drawings of clustered graphs. Tech. Report RT-DIA-144-2009, Dept. , Roma Tre Univ. pdf 2. : Completely connected clustered graphs. J. Discr. Alg. 4(2), 313–323 (2006) 3. : C-planarity of c-connected clustered graphs. J. Graph Alg. Appl. 12(2), 225–262 (2008) 4.

Let d denote the size of the s-set of smallest cardinality that was inserted in round j − 1. For every i ∈ [1, 2j − 1], we insert l vertices at depth i/2j , one for each column in E, unless some other w vertex has been inserted at this same depth in a previous round, in which case we do not perform any additional insertion. This has the eﬀect that 2j−1 additional layers of vertices are inserted in round j. We call the i-th deepest layer of vertices that is inserted in round j the i-th layer of round j, and denote by wj,i,1 .