View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 308 (2008) 6596–6600 www.elsevier.com/locate/disc Note A characterization of planar partial cubes Iztok Peterin Institute of Mathematics and Physics, FEECS, University of Maribor, Smetanova ulica 17, 2000 Maribor, Slovenia Received 17 March 2006; received in revised form 27 November 2007; accepted 29 November 2007 Available online 4 January 2008 Abstract Partial cubes as well as planar graphs have been extensively investigated. In this note we introduce an additional topological kind of condition to the Chepoi’s expansion procedure that characterizes planar partial cubes. As a consequence we obtain a characterization of some other planar subclasses of partial cubes. c 2007 Elsevier B.V. All rights reserved. Keywords: Partial cubes; Planar graphs; Expansion 1. Introduction and preliminaries Partial cubes are isometric subgraphs of hypercubes and have been largely investigated, see the book [8] and the references therein. The most important subclass of partial cubes are median graphs. Both classes are precisely determined with some expansion procedure. That is, any partial cube can be obtained from K1 by a certain sequence of graph enlargements as shown by Chepoi [4]. The same holds for median graphs (only the rule is different) as proved by Mulder [10,11]. In [12] a topological kind of condition was introduced that ensures – together with Mulder’s expansion theorem – planarity of median graphs. A natural question arose whether a similar condition exists for planar partial cubes. Here we introduce such a condition, that is even more natural than the one in [12]. Surprisingly it gives a characterization – together with Chepoi’s expansion theorem – of planar partial cubes. The same condition also holds for graph classes that lie between median graphs and partial cubes and can be obtained from K1 by (some) expansion. For additional information on these classes of graphs we recommend [3]. In the remainder of this section we fix the notation. In the next section the main result follows and the discussion of planarity for other graph classes that can be obtained by some expansion procedure. The distance dG .u; v/ between two vertices u and v in a graph G is defined as the number of edges on a shortest u; v-path. A subgraph H of G is called isometric, if dH .u; v/ D dG .u; v/ for all u; v 2 V .H/ and H is convex if for every u; v 2 V .H/ all shortest u; v-paths belong to H. Convex subgraphs are clearly isometric. The Cartesian product GH of two graphs G and H is the graph with vertex set V .G/ × V .H/ where the vertex .a; x/ is adjacent to .b; y/ whenever ab 2 E.G/ and x D y, or a D b and xy 2 E.H/. Hypercubes or n-cubes Qn E-mail address: [email protected]. 0012-365X/$ - see front matter c 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2007.11.066 I. Peterin / Discrete Mathematics 308 (2008) 6596–6600 6597 are Cartesian products of n copies of K2. Isometric subgraphs of hypercubes are called partial cubes. Trees and even cycles are partial cubes. Let G1 and G2 be two isometric subgraphs of a graph G that form a cover of G with nonempty intersection G1 \ G2 D G0. Note that there is no edge from G1nG0 to G2nG0. Graph H is an expansion of G with respect to G1 and G2 as follows. Take disjoint copies of G1 and G2 and connect every vertex from G0 in G1 with the same vertex of G0 in G2 with an edge. Such pairs of vertices will be called expansion neighbors. We say that expansion is isometric (connected) if G0 is isometric (connected). It is not hard to see that copies of G0 in G1 and in G2 and new 0 edges between those two copies form the Cartesian product G K2. In [4] Chepoi has shown that G is a partial cube if and only if it can be obtained from K1 by a sequence of expansions. One of the most useful relations for the investigation of metric properties of graphs in general and partial cubes in particular is the Djokovic–Winkler´ relation Θ, (cf. [5,13]). Two edges e D xy and f D uv of G are in the relation Θ if dG .x; u/ C dG .y; v/ 6D dG .x; v/ C dG .y; u/: Clearly, Θ is reflexive and symmetric, but not transitive in general. Winkler proved in [13] that transitivity of Θ is characteristic for partial cubes among bipartite graphs. Graph G is planar if it can be drawn in the plane such that any two edges cross only in an endvertex (if they are incident with the same endvertex). Such drawings are called plane drawings of G. Any plane drawing of G divides the plane into regions which are called faces. One of those faces is unbounded and is called the exterior or the outer face, the others are interior or inner faces. Vertices that lie on an outer face are called outer vertices and others are inner vertices. Note that the boundary of every face of some plane drawing can be the boundary of an outer face of some other plane drawing of the same graph. A graph G is outer planar if it is planar and embeddable into the plane so that all vertices lie on the outer face of the embedding. In [1] Behzad and Mahmoodian have shown that G is outer planar if and only if GK2 is planar. For more information on planar graphs (or more general graphs on surfaces) we recommend [9]. 2. Two-face expansions Vertex u of a graph G is a cut-vertex if G − u has more components than G, while edge e is a bridge if G − e has more components than G. (We remove only the edge e without endvertices.) Let G be a planar graph. We construct the graph G− as follows. First delete all bridges from G. Let u be a cut- vertex in the obtained graph. We delete u, add copies of u back to all components incident with u in the natural way − − and denote this graph with Gu . With G we denote the graph that remains from G after this procedure is executed for all cut-vertices of G. For a tree T on n vertices we get the totally disconnected graph on n vertices for T − and if − G is obtained by amalgamating a vertex from cycle Cn with a vertex of cycle Cm, then G consists of disjoint cycles Cn and Cm. Let H be an expansion of a planar graph G with respect to G1 and G2. Then H is a 2-face expansion of G if all vertices of G0 D G1 \ G2 are on one face of some plane drawing of G1 and on one face of some plane drawing of G2. First we need two technical lemmas. Lemma 1. Let G be a planar 2-connected graph with a subdivision S of K2;3 and fix planar drawing D. Suppose that there exist vertices u1; u2, and u3 of S that lie pairwise on the same face of D but not all three on the same face. 0 0 Then there exists a subdivision S of K2;3 where fu1; u2; u3g is one part of a partition of S . Proof. Let fv1; v2g and fw1; w2; w3g be the sets that form a partition of S. Let P1, P2, and P3 be the v1; v2-paths from S. If fw1; w2; w3g D fu1; u2; u3g, there is nothing to prove. Thus first suppose that fu1; u2; u3g 2 S and that they do not form one set of a partition of S. If all ui ’s lie on one v1; v2-path, say P1, they have two common faces in S. Suppose that u1 is closest to v1 on P1, u3 is closest to v2 on P1, and u2 in between. To ensure that fu1; u2; u3g do not all lie on the same face there must be at least one additional path in G. If there exists a x1; x2-path where x1 is on u1; u2-subpath of P1 and x2 is not on u2; u3-subpath of P1. Then there must also exists a u1; x3-path where x3 is on u2; u3-subpath of P1, otherwise we have a contradiction with the 6598 I. Peterin / Discrete Mathematics 308 (2008) 6596–6600 0 assumptions. But then fu1; u2; u3g and fx1; x3g form a partition of a subdivision S of K2;3. (The case when x1 is on u2; u3-subpath of P1 and x2 is not on u1; u2-subpath of P1 is symmetric.) If there exists a x1; x2-path where x1 is on u1; u2-subpath of P1 and x2 is on u2; u3-subpath of P1. Then there must also exists a u2; x3-path where x3 is on w1; u1-subpath of P1 or on u3; w2-subpath of P1, otherwise we have a contradiction with the assumptions. Note that x1 can be u1 and x2 can be u3, but not both at the same time. Suppose 0 that x3 6D u3. Again fu1; u2; u3g and fx1; x3g form a partition of a subdivision S of K2;3. Now let two vertices, say u1 and u2, be on P1 and u3 on P2.