Graphs with Forbidden Subgraphs

Many graphs which are encountered in the study of graph theory are characterized by a type of configuration or subgraph they possess. However, there are occasions when such graphs are more easily defined or described by the kind of subgraphs they are not permitted to contain. For example, a tree can be defined as a connected graph which contains no cycles, and Kuratowski [22] characterized planar graphs as those graphs which fail to contain subgraphs homeomorphic from the complete graph KS or the complete bipartite graph K3.s . The purpose of this article is to study, in a unified manner, several classes of graphs, which can be defined in terms of the kinds of subgraphs they do not contain, and to investigate related concepts. In the process of doing this, we show that many “apparently unrelated” results in the literature of graph theory are closely related. Several unsolved problems and conjectures are also presented. DEFINITIONS AND NOTATION Before beginning our study, we present definitions of a few basic terms and establish some of the notation that will be employed throughout the article. Definitions not given here may be found in [18]. *Research supported in part by a grant from the National Science Foundation (GN-2544). 12 GRAPHS WITH FORBIDDEN SUBGRAPHS 13 The graphs under consideration are ordinary graphs, i.e., finite undirected graphs possessing no loops or multiple lines. The points of a graph G are usually denoted by u, u, w and the lines by x, y, z. If x joins the points u and v, then we write x = uv. The degree of a point u in a graph G is denoted deg u. The smallest degree among the points of G is denoted min deg G while the largest such number is max deg G. A subgraph H of a graph G consists of a subset of the point-set of G and a subset of the line-set of G which together form a graph. Two special but important types of subgraphs are the following. The subgraph induced by a set U of points of G has U for its point-set and contains all lines of G incident with two points of U. The subgraph induced by a set Y of lines of G has Y for its line-set and contains all points incident with at least one line of Y. Two subgraphs are disjoint if they have no points in common and line-disjoint if they have no line in common. A connected component of a graph G is a maximal connected subgraph of G. A cut-point of a connected graph G is a point whose removal disconnects G. A bridge is a line whose removal disconnects G. A block of G is a maximal connected subgraph of G containing no cut-points. The connected components of G partition its point-set while the blocks of G partition its line-set. Two important classes of graphs are the complete graphs and the complete bipartite graphs. The complete graph K, has each two of its p points adjacent. The complete bipartite graph K,,, or K(m, n) has m + II points; and its point-set can be partitioned into two subsets, one containing m points and the other n points, such that each point of one subset is adjacent with every point of the other subset but no two points in the same subset are adjacent with each other. In general, then, the complete npartite graph K(pl , pz ,.,., pn) has Zpi points and its point-set can be partitioned into subsets Vi, 1 < i < n, such that I Vi / = pi and two points u and v are adjacent if and only if u E Vj and v E VI< , where j f k. A subdivision of a graph His a graph G1 obtained from H by replacing some line x = uv of H by a new point w together with the lines uw and VW. A graph G is then said to be homeomorphic from a graph H if G can be obtained from H by a finite sequence of subdivisions. Two graphs Gr and G, are homeomorphic with each other if there exists a graph G, homeomorphic from both G, and G, . GRAPHS WITH PROPERTY P, For a real number r, [r] and {r} denote the largest integer not exceeding r and the least integer not less than r, respectively. We say that a graph G 14 CHARTRAND, GELLER AND HEDETNIEMI has property P, , where n is a positive integer, if G contains no subgraph which is homeomorphic from the complete graph K,,, or the complete bipartite graph A few observations now follow readily from this definition. A totally disconnected graph is one which has no lines. A graph which contains no subgraph homeomorphic from K, or K,,, necessarily has no lines. (The graph K,,, is actually superfluous here since it is homeomorphic from K, .) Thus, a graph has property PI if and only if it is totally disconnected. A forest is a graph without cycles. If a graph contains no subgraph homeomorphic from K3 or Kzs2 , it is forbidden to contain cycles. (Here again K,,, itself is homeomorphic from K3 .) Hence the graphs with property P, are the forests. An outerplanar graph is a graph G which can be embedded in the plane so that every point of G lies on the exterior region. Tang [27] has investigated several properties of outerplanar graphs, while Chartrand and Harary [ 121 have characterized outerplanar graphs as those graphs which fail to contain subgraphs homeomorphic from K4 or Kz,3 . Therefore, the graphs with property P, are the outerplanar graphs. Whenever an outerplanar graph is encountered in this article, we shall assume it is embedded in the plane so that all of its points lie on the exterior region. A planar graph is one which can be embedded in the plane. The wellknown theorem of Kuratowski [22] states that a graph is planar if and only if it contains no subgraph homeomorphic from K5 or K3,3 . Hence, the graphs with property Pa are the planar graphs. Throughout this article it is assumed all planar graphs are embedded in the plane. Thus far, no special name has been given to graphs having property P, , where n > 5. In this article, several results dealing with graphs having property P, , 1 < n < 4, are given. The similarity in these results lead to natural conjectures, which are also presented. It is no surprise that the definition of graphs with property P, was inspired by Kuratowski’s characterization of planar graphs. Another well-known characterization of planar graphs with an amazing resemblance to Kuratowski’s theorem involves contractions. One might well wonder if the definition of property P, could be given in an equivalent form using contractions. We now consider this question. A graph G’ is said to be a contraction of a graph G if there exists a oneto-one correspondence between the point-set of G’ and the subsets determined by a partition of the point-set of G such that each of these subsets GRAPHS WITH FORBIDDEN SUBGRAPHS 15 induces a connected subgraph of G and such that two points of G’ are adjacent if and only if there is a line joining points of the corresponding subsets. Let G’ be a contraction of the graph G. The subgraph induced by a set of lines of G’ is called a subcontraction of G. In [28] it is shown that any subcontraction of G can be realized by a contraction of a subgraph of G induced by a set of lines. It is known (see [19]) that homeomorphism is a special case of subcontraction, i.e., if a graph G contains a subgraph which is homeomorphic from a graph H, then H is a subcontraction of G. This, however, implies the following: PROPOSITION 1. If a graph G has neither K,,, nor as a subcontraction, then G has property P, . The converse of Proposition 1 is known to hold for n = 1, 2, 3,4. Indeed, the following results state precisely this fact. We present these in our terminology. THEOREM (Hahn [17]). For 1 < n < 3, a graph G has property P, if and only if G has neither K,,, nor