A Note on IC-Planar Graphs
AA Note on IC -Planar Graphs Christian Bachmaier, Franz J. Brandenburg, and Kathrin Hanauer
University of Passau, 94030 Passau, Germany { bachmaier|brandenb|hanauer } @fim.uni-passau.de Abstract.
A graph is IC -planar if it admits a drawing in the plane withat most one crossing per edge and such that two pairs of crossing edgesshare no common end vertex. IC -planarity specializes both NIC -planarity,which allows a pair of crossing edges to share at most one vertex, and -planarity, where each edge may be crossed at most once.We show that there are infinitely maximal IC -planar graphs with n ver-tices and 3 n − A graph G is maximal in a graph class G if no edge can be added to G withoutviolating the defining class. The density ( sparsity ) of G is an upper (lower) boundon the number of edges of maximal graphs G ∈ G with n vertices. A maximalgraph G is densest ( sparsest ) in G if its number of edges meets the upper (lower)bound.It is well-known that every maximal planar graph is triangulated and has3 n − -planar graphs, which are graphs that can be drawn with at mostone crossing per edge. These graphs have recently received much interest [13]. -planar graphs with n vertices have at most 4 n − n = 8 and all n ≥
10 [5, 7]. However, there are sparse maximal -planar graphs with less than 2 . n edges [10]. The best known lower bound onthe sparsity of -planar graphs is 2 . n [5] and neither the upper nor the lowerbound are known to be tight.There are some subclasses of -planar graphs with different bounds for thedensity and sparsity. A graph is IC -planar (independent crossing planar) [1,8,14,15] if it admits a drawing with at most one crossing per edge so that each vertexis incident to at most one crossing edge, and NIC -planar (near independentcrossing planar) if two pairs of crossing edges share at most one vertex [15]. IC -planar graphs have an upper bound of 3 . n − n = 4 k and k ≥ NIC -planar graphs are ( n −
2) [15] and ( n −
2) [3,11]and both bounds are known to be tight for infinitely many values of n . Outer -planar graphs are another subclass of -planar graphs. They must admit a a r X i v : . [ c s . D M ] J u l C. Bachmaier, F. J. Brandenburg, K. Hanauer
Table 1: The density of maximal graphs on n vertices. NIC -planar IC -planar outer 1-planarupper bound 4 n − ( n −
2) [3, 16] n − n − (cid:98) example 4 n − ( n −
2) [3, 11] n − n − n − [4] ( n −
2) [3] 3 n − n − [2] (cid:98) example n − [9] ( n −
2) [3] 3 n − n − [2] -planar embedding such that all vertices are in the outer face [2,12]. Results onthe density of maximal graphs are summarized in Table 1.Here, we establish a lower bound of 3 n − IC -planar graphsand show that it is tight for all n ≥ We first prove the existence of maximal IC -planar graphs that have n verticesand only 3 n − Lemma 1.
For every n ≥ there is a maximal IC -planar graph with n verticesand n − edges.Proof. As K has exactly 3 n − IC -planar, the statementtrivially follows for n = 5. Let us hence assume in the following that n ≥
6. Weconstruct a graph G n with n vertices and 3 n − G n consists of n − C = ( v , v , . . . , v n − ) as well as two pole vertices p and q . For every 0 ≤ i < n − G n has edges { v i , p } and { v j , q } . Additionally,there is an edge { p, q } connecting the poles. As an example, Fig. 1a depicts thegraph G . Then, every vertex v i , 0 ≤ i < n −
2, is incident to exactly two circleedges as well as to both p and q , and p and q are each incident to n − G n has (4( n −
2) + 2( n − n − n − E ( G n ) be any IC -planar embedding of G n . We will now show that { p, q } must be crossed in E ( G n ) and that E ( G n ) is unique up to isomorphism.Suppose that an edge { v i , v i +1 } crosses an edge { v j , v j +1 } in E ( G n ) (seeFig. 1b). Due to IC -planarity, { v i , p } , { v i +1 , p } , { v j , q } , and { v j +1 , q } must beplanar. In consequence of the crossing, v j and v j +1 lie on different sides of theclosed path P consisting of { v i , p } , { v i , v i +1 } , and { v i +1 , p } . Hence, either { v j , q } or { v j +1 , q } must cross an edge of P , a contradiction. Thus, every crossing in E ( G n ) must involve at least one of p or q .Suppose that an edge { v i , p } crosses an edge { v j , v j +1 } in E ( G n ) (see Fig. 1c).By IC -planarity, { v j , q } and { v j +1 , q } must be planar. As n ≥
6, there must be avertex v i (cid:48) adjacent to v i . Furthermore, { v i (cid:48) , v i } and { v i (cid:48) , p } must be planar dueto IC -planarity. In consequence of the crossing, p and v i however lie on differentsides of the closed path P consisting of { v j , q } , { v j , v j +1 } , and { v j +1 , q } , so either Note on IC -Planar Graphs 3 pq v v v v v v v i v i +1 v j v j +1 p qp v i v j v j +1 qv i (cid:48) pv i q v j v i (cid:48) v j (cid:48) (a) (b)(c) (d) Fig. 1: Proof of Lemma 1: The graph G n for n = 8 (a) along with sketches ofthe cases where an edge { v i , v i +1 } crosses an edge { v j , v j +1 } (b), { v i , p } crossesan edge { v j , v j +1 } (c), and { v i , p } crosses an edge { v j , q } (d), which all yieldnon- IC -planar embeddings of G n . { v i (cid:48) , v i } or { v i (cid:48) , p } must cross an edge of P , a contradiction. Thus, every crossingin E ( G n ) must involve both p and q .Finally, suppose that an edge { v i , p } crosses an edge { v j , q } in E ( G n ) (seeFig. 1d). As n ≥
6, there must be two further vertices v i (cid:48) (cid:54) = v j (cid:48) such that v i (cid:48) isadjacent to v i and v j (cid:48) is adjacent to v j . Furthermore, { v i , v i (cid:48) } , { v j , v j (cid:48) } , { v i (cid:48) , p } ,and { v j (cid:48) , q } must be planar due to IC -planarity. In result of the crossing, v j and q lie on different sides of the closed path P consisting of { v i , p } , { v i , v i (cid:48) } , and { v i (cid:48) , p } . Thus, one of { v j , v j (cid:48) } and { v j (cid:48) , q } must cross P , a contradiction.Consequently, every crossing in E ( G n ) must contain { p, q } , which in turn canonly cross an edge { v i , v i +1 } for some i with 0 ≤ i < n − { v n − , v } , asdepicted, e. g., in Fig. 1a. As no edge can be added to G n and E ( G n ) such that IC -planarity is preserved, G n is maximal. (cid:117)(cid:116) Note that the embedding of the graphs G n from the proof of Lemma 1 areunique up to isomorphism, because { p, q } must cross an arbitrary edge { v i , v i +1 } .Concerning the upper bound, observe that every maximal IC -planar graph with n ≥ n − C. Bachmaier, F. J. Brandenburg, K. Hanauer
Theorem 1.