The genus of the 2‐amalgamations of graphs

Abstract

A graph G is called the 2‐amalgamation of subgraphs G1 and G2 if G = G1 ∪ G2 and G1 ∩ G2 = {x, y}, 2 distinct points. in this case we write G = G1∪{x, y} G2. in this paper we show that the orientable genus, γ(G), satisfies the inequalities γ(G1) + γ(G2) − 1 ≤ γ(G1 ∪{x, y} G2) ≤ γ(G1) + γ(G2) + 1 and that this is the best possible result, i. e., the resulting three values for γ(G1 ∪{x, y} G2) which are possible can actually be realized by appropriate choices for G1 and G2.

Additivity of the crossing number of graphs with connectivity 2

“…It is quite natural to ask about conditions which would guarantee equality in (1). A simple such condition is given in the following theorem.…”

Obstructions of Connectivity Two for Embedding Graphs into the Torus

“…Let (G) = 1 if G is xy-alternating and (G) = 0 otherwise. 3 We shall also use the graph parameter + defined as + (G) = (G + ).…”

“…A graph G is a k-sum of graphs G 1 = (V 1 , E 1 ), G 2 = (V 2 , E 2 ) if G can be written as G = (V 1 ∪V 2 , E 1 ∪E 2 ) such that |V 1 ∩V 2 | = k. It is easy to show that obstructions that are not 2-connected can be obtained as disjoint unions and 1-sums of obstructions for surfaces of smaller genus (see [1]). Stahl [10] and Decker et al [3] showed that the genus of a 2-sum differs by at most 1 from the sum of genera of its parts. Decker et al [4] provided a simple formula for the genus of a 2-sum that will be used in this paper.…”

Obstructions for two-vertex alternating embeddings of graphs in surfaces

“…Given graphs G 1 and G 2 such that V (G 1 ) ∩ V (G 2 ) = {x, y}, the union of G 1 and G 2 , that is the graph (V (G 1 ) ∪ V (G 2 ), E(G 1 ) ∪ E(G 2 )), is an xy-sum of G 1 and G 2 (or a 2-sum if the vertices are not important). Sometimes, we also call G to be an xy-sum of G 1 and G 2 even if the edge xy is an edge of G 1 or G 2 but is not present in G. To determine the genus of the xy-sum of G 1 and G 2 , it is neccessary to know if G 1 (and G 2 ) has a minimum genus embedding Π such that there is a Π-face in which x and y appear twice in the alternating order (see [4,5]). For vertices x, y ∈ V (G), we say that G is xy-alternating on S k if g(G) = k and G has an embedding Π of genus k with a Π-face W = v 1 .…”