Amalgamation of a graph

Amalgamation of a graph — амальгамация графа.

Amalgamating a graph [math]\displaystyle{ \,H }[/math] can be thought of as taking [math]\displaystyle{ \,H }[/math], partitioning its vertices, then, for each element of the partition, squashing together the vertices to form a single vertex in the amalgamated graph [math]\displaystyle{ \,G }[/math]. Any edges incident with original vertices in [math]\displaystyle{ \,H }[/math] are then incident with the corresponding new vertex in [math]\displaystyle{ \,G }[/math], and any edge joining two vertices that are squashed together in [math]\displaystyle{ \,H }[/math] becomes a loop on the new vertex in [math]\displaystyle{ \,G }[/math]. The number of vertices squashed together to form a new vertex [math]\displaystyle{ \,w }[/math] is the amalgamation number [math]\displaystyle{ \,\eta(w) }[/math] of [math]\displaystyle{ \,w }[/math]. The resulting graph is the amalgamation of the original. Formally, this is represented by a graph homomorphism [math]\displaystyle{ f: V(G) \rightarrow V(H) }[/math]; so for example if [math]\displaystyle{ w \in V(H) }[/math], then [math]\displaystyle{ \,\eta(w) = |f^{-1}(w)| }[/math].