Finding All Chordless Simple Cycles In A Graph

I am doing a research on networks which consists of polygons with different number of sides. I am trying to find all simple cycles in a network which are chordless. As an example, consider the following graph:

graph = Graph[
  {
    1 <-> 2,   2 <-> 3,   3 <-> 4,  4 <-> 5,   5 <-> 6,   1 <-> 6,  
    3 <-> 7,   7 <-> 8,   8 <-> 9,  9 <-> 10, 10 <-> 11, 11 <-> 3,
    4 <-> 12, 12 <-> 13, 13 <-> 11
  }, 
    VertexLabels -> "Name"
]

{1,2,3,4,5,6}, {3,4,11,12,13},{3,7,8,9,10,11} are rings and we can extract them:

cycles = FindFundamentalCycles[graph];
rings = Sort @* VertexList @* Graph /@ cycles  

But the above solution doesn't always work as it might give non-chordless cycles. Consider the following example:

grapht = Graph[
{

  1 <-> 2, 1 <-> 3, 2 <-> 4, 4 <-> 5, 5 <-> 6, 4 <-> 6, 
  6 <-> 7, 3 <-> 5, 3 <-> 9, 5 <-> 8, 8 <-> 9
},
VertexLabels -> "Name"];

Rings (cycles) are:

cyclest = FindFundamentalCycles[grapht]; 
HighlightGraph[grapht, #] & /@ cyclest

But I need to get {4,5,6} as a ring not {1,2,3,4,5,6} since there is an edge in the latter. Is there any way to filter out only chordless cycles?

---

W Community crosspost

Answer

First, let us find all cycles in the graph. Then, we will filter out the ones that contain chords; this we can detect by checking if the $n$-vertex induced subgraph is isomorphic to a cycle of length $n$ or not.

Let us use your graph as an example:

g = Graph[{1 <-> 2, 1 <-> 3, 2 <-> 4, 4 <-> 5, 5 <-> 6, 4 <-> 6, 
   6 <-> 7, 3 <-> 5, 3 <-> 9, 5 <-> 8, 8 <-> 9}, 
  VertexLabels -> "Name"];


cy = VertexList[Graph[#]] & /@ FindCycle[g, Infinity, All];
Select[cy, IsomorphicGraphQ[CycleGraph[Length[#]], Subgraph[g, #]] &]
(* {{4, 5, 6}, {5, 8, 9, 3}, {1, 2, 4, 5, 3}} *)

HighlightGraph[g, %]

Of course, if you have additional constraints, you can simply modify the call to FindCycle with different parameters to only find cycles of length e.g. of size 5, 6, 7, or 8. To achieve this, just do FindCycle[g, {5, 8}, All] instead.