Compositions of Unions of Graphs

The concept of a composition of a graph was introduced by Arnold Knopfmacher and M.E. Mays in [2] and denotes a partition of the vertices such that the induced subgraph on each part is connected. Alternatively, a partition can be viewed as the set of connected components of a subgraph including all the vertices of the graph but only a subset of the edges. Most attention hitherto has been devoted to counting the numbers of compositions of various families of graphs, usually by finding some ad hoc recurrence relation based on their structure. We shall develop a more systematic method of analysis of how the composition of the union of two graphs can be obtained from the compositions of the two subgraphs. This provides new proofs giving greater insight into some known results (ladder graphs and wheel graphs), a new result on 3×n grid graphs, and a general result for the cartesian product of an arbitrary graph and a path with n vertices. All graphs considered will be finite and undirected, with no loops or multiple edges. If G is a graph, then V (G) denotes its vertex set and E(G) denotes its edge set, where each edge can be thought of as an unordered pair of vertices. We shall, however, also use the more symbolic notation x–y to denote the edge between vertices x and y. The cartesian product A × B has vertex set V (A) × V (B), and {(a1, b1), (a2, b2)} is an edge if and only if a1 = a2 and {b1, b2} ∈ V (B), or {a1, a2} ∈ V (A) and b1 = b2. Since any composition of the graph is a partition of the vertex set, it defines an equivalence relation. Vertices x and y that are in the same part in a given composition will therefore simply be said to be related, written x ∼ y. It is convenient to define a distance function associated with any composition of a connected graph G. If x and y are vertices of G, then d(x, y) denotes the minimum number of additional edges of G that must be incorporated into the composition in order for x and y to become related (that is, d(x, y) is the length of the shortest path in G joining their equivalence classes). For example, if x ∼ y, then d(x, y) = 0, and if x and y are adjacent in G but x 6∼ y, then d(x, y) = 1. In general, more than one subset of E(G) may define the same composition of G. (For example, with the cycle Cn having n vertices and n edges, E(G) obviously defines the composition with only one part, as does any subset of E(G) containing n − 1 edges.) However, we shall say that an edge x–y belongs to a particular composition if its two endpoints are in the same part, that is, if x ∼ y. Thus the edges belonging to a given composition of G form the largest subset of E(G) defining that composition. The number of compositions of a graph G will be denoted by C(G). Suppose graph G = A ∪ B, where E(A) ∩ E(B) = ∅, so A ∩ B is a null graph. The problem is to determine C(G) from C(A) and C(B). If Pn denotes the path with n vertices, then C(A × Pn) can be