# composition of graphs while eliminating inner edges

I am having left side sub graphs and want to compose each after the other. my objective is to get the right side graph.

question 1: can any one suggest me a way to do this.

(Note: I am having cycles, and as the cycles can be considered as the faces of a planar graph, we can consider this as planar graph as well. not sure whether this is relevant to here or not)

so, looking for a operational process to get my final graph

Question 2: Does my final graph that i am looking for be the outer face?

thank you,

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I find it a bit unclear what you are asking, but a quick look tells me that the final graph that you want is the symmetric difference of all the graphs on the left. Equivalently, if you take the adjacency matrices of all the graphs on the left (with vertex set $\{ 1 \dots 11 \}$) and add them up $\mod 2$, you should get the adjacency matrix of the graph on the right. –  Andrew Salmon Apr 7 '13 at 4:51
@Andrew Salmon: can i get my final graph by getting difference of union and intersection of adjacent graphs one after the other. because i should avoid all inner edges like 2,3 & 7,8. at the end i think this graph is the outer face of planar graph. is that? –  niro Apr 7 '13 at 8:01