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Prove that for a graph, that has it's each edge belonging to a cycle, can't be...

Question:

Prove that for a graph, that has its each edge belonging to a cycle, cant be reconstructed, only from its cycle matur, cyc
Prove that for a graph, that has it's each edge belonging to a cycle, can't be reconstructed, only from it's cycle matur, cycle matrix: let graph 6 have medges and let q be the number of different cycles in G. The cycle matrix B= [ bij]gxm of 6 is à (0,1) matrix of order qxm, with big-1. if the 'irth cycle includes j-th edge. O otherurse

Answers

Consider a graph having each of its vertex belonging to a certain cycle. Consider the case when there are shared edges between the cycles. We can construct more that one graph in when a cycle shares it's two different edges with two different cycles. Hence, we can con construct a unique graph in every case given cycle matrix. For example,  we have information from cycle matrix that a particular edge being part which all cycles.

Consider the case when we have a graph with edges { a,b,c,d,e,f,g,h,i, j,k,x,y}, let edge {a,b,c,d,x} belongs to cycle 1, let {x,e,f,g,h,y} belong to cycle 2 and {y,i,j,k} belong to cycle 3. In this case, we can construct 2 graphs, both having the same cycle matrix:

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