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graphs and networks - Combinatorica: Girth[] and FindCycle[] disagreement



Warning: run the following code in a fresh Mma session, as some symbols could be shadowed (depending on your Mma version)


While trying to answer this question, I fell into the following:


 (* Let's load a large Directed Graph and convert it to Combinatorica *)
g = Graph@Union@Flatten[
     Thread[DirectedEdge @@ ##] & /@ Select[{#, IsotopeData[#, "DaughterNuclides"]} & /@ 
        IsotopeData[], #[[2]] != {} &]];
Needs["GraphUtilities`"]
<< Combinatorica`
cg = ToCombinatoricaGraph[g];



Girth[cg] gives the length of a shortest cycle in a simple graph g.



So, let's check if cg is Simple and calculate its Girth:


{SimpleQ@cg, Girth@cg}
(*
-> {True, 3}
*)

So there is at least one Cycle in cg of length 3.



But look what happens when we try to find it by the two available methods in Combinatorica:


{ExtractCycles@cg, FindCycle@cg}
(*
-> {{},{}}
*)

So, two questions:



  1. Is this a bug?

  2. What is the easiest way to find all cycles in g without using Combinatorica?



Edit


BTW, the (now) standard Graph functionality also detects cycles:


AcyclicGraphQ[g]
(* -> False *)

Answer



I don't think it's a bug. Directed radioactive decay graphs shouldn't have cycles by definition and AcyclicGraphQ doesn't see them either:


AcyclicGraphQ@g



True



You don't seem to have heeded your own warning to start with a fresh kernel, as playing around with the code for a while gave me False too.


The output of ExtractCycles and FindCycle is therefore correct.


Girth doesn't seem to take directionality into account when determining cycles:


Cyclic directed graph:


Girth@ ToCombinatoricaGraph@
     System`Graph[{1 \[DirectedEdge] 2, 2 \[DirectedEdge] 3, 3 \[DirectedEdge] 1}]



3



Acyclic directed graph (note the reversed direction of the last edge):


Girth@ToCombinatoricaGraph@
     System`Graph[{1 \[DirectedEdge] 2, 2 \[DirectedEdge] 3, 1 \[DirectedEdge] 3}]


3



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