Question

2. Find a chordal graph for which the degeneracy ordering (the ordering obtained by repeatedly deleting a vertex of minimum degree) is not an elimination ordering (an ordering in which every vertex and its later neighbors form a clique). If your graph has more than one degeneracy ordering then none of its degeneracy orderings should be elimination orderings. (Because your graph is chordal, it should have at least one elimination ordering. Hint: Choose your graph so that there is only one possible starting vertex for the degeneracy ordering, and so that the neighbors of that vertex do not form a clique.)

Answer 1

Consider the graph shown in figure below, in which the minimum degree vertices are {1,2,3} each of degree 1 and then the vertices {4,5,6}. Clearly this graph is a chordal graph since there is only one cycle which is of length 3.

2. 6 4 3

Any degeneracy order will at first remove the vertices {1,2,3} in any order it does not matter and we can see that {1,2,3} are not neighbours of each other. Hence degeneracy order will first eliminate vertices {1,2,3} and then {4,5,6} and since vertices {1,2,3} in any order are not neighbours of each other, hence any degeneracy order in this graph will not be elimination order.

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