... q) planar graph of degree k and f regions with l boundaries, then (k – 2)(l − 2) < 4. Hint: Let G be any planar graph with p vertices, q edges and r faces. If the face length is l and the Deg(G) = k, then we have pk = ln = 2q. Again, q ...
... q) graph with r regions, then p − q + r = 2. Proof. We proceed by induction on the size of the graph. If q = 0, then since G is connected, p = 1 and r = 1, and the result follows. Now, assume the result is true for all connected plane ...
... ( Q ) . Let J ' be the subgraph of G induced by the vertices PG ( P , v ) U set ( Q ) without the edges between ... graph from M is chosen in a variable H3 and a new process RECOGNIZE with the parameter list ( H3 , Q , w ) is started for each w ...
... graph. To formalize this idea, let us first define a Q-graph as a labeled undirected graph γ = 〈V,E,L〉, where V is a finite set of nodes, E ⊆ V ×V \{〈v, v〉 | v ∈ V} is a finite set of edges, and L is a labeling function from V to ...
... q) graph is a tree iff it is a acyclic and p = q + 1, or q = p – 1 (Theorem 2.2). Conversely, we assume G is connected (p, q) graphs with p = q + 1 It is sufficient to show that G is acyclic. If G contains a cycle C and e is an edge of ...
... graphs with q = 2p - 3 . 1. Introduction By a graph we mean a finite , undirected , connected graph without loops or multiple edges . We denote by p and q the order and size of G. Terms not defined here are used in the sense of Harary ...
Frank Harary. p−q+r=2. (11.1′) It is easy to prove this theorem by induction. However, this equation has already been proved in Chapter 4 where it was established that the cycle rank m of a connected graph G is given by m=q−p+1. Since ...
... graph with n ver- tices , q edges , and r regions , then n - q + r = 2 . = 0 , then G must be K1 , a Proof . We induct on q , the number of edges . If q graph with 1 vertex and 1 region . The result holds in this case . Assume that the ...
... graph is a pentagon , and the unique connected locally pentagon graph is the icosahedron . For q = 9 the Paley graph is the 3 x 3 grid , and there are precisely two connected locally 3 x 3 graphs , namely the Johnson graph ( 3 ) on 20 ...
... graph H is a minor of graph G, then μ(H) ≤ μ(G). D tite Example graph Kp,q 16.10. by adding Let K+ all p,q edges denote in the the color graph class obtained of size from p. We the are complete interested biparin the case q ≥ 3. nodes ...