How do I "read out" the vertex names on this graph?

Question

While working to self-answer FindCurvePath for lines (rather than points) using Graph functions I came upon a seemingly silly conundrum: how do I programmatically list the vertex names of a PathGraph-like-graph in order? For example I have:

How can I extract {9, 10, 1, 2, 4, ..., 41, 34, 33} from this?

I am using version 10.1.0 and I do not have FindHamiltonianPath as proposed by Jason. I suppose that is the canonical approach in recent versions.

Code:

Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 
  22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 
  42, 43, 44}, {Null, 
  SparseArray[Automatic, {44, 44}, 
   0, {1, {{0, 2, 4, 6, 8, 10, 12, 14, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 
      37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 
      74, 76, 78, 80, 82, 84, 
      86}, {{2}, {10}, {1}, {4}, {4}, {8}, {2}, {3}, {6}, {24}, {5}, {7}, {6}, {8}, \
{3}, {7}, {10}, {1}, {9}, {12}, {20}, {11}, {16}, {14}, {15}, {13}, {18}, {13}, \
{16}, {12}, {15}, {18}, {22}, {14}, {17}, {20}, {42}, {11}, {19}, {22}, {28}, {17}, \
{21}, {24}, {25}, {5}, {23}, {23}, {26}, {25}, {29}, {28}, {32}, {21}, {27}, {26}, \
{30}, {29}, {37}, {32}, {36}, {27}, {31}, {34}, {33}, {41}, {36}, {44}, {31}, {35}, \
{30}, {38}, {37}, {39}, {38}, {40}, {39}, {43}, {34}, {42}, {19}, {41}, {40}, {44}, \
{35}, {43}}}, Pattern}]}, {VertexLabels -> {"Name"}}]

Answer 1

We can use GraphPeriphery to get the start and endpoints, and then use these in FindPath:

periphery = GraphPeriphery[g]

FindPath[g, ##]& @@ periphery

{9,33}

{{9,10,1,2,4,3,8,7,6,5,24,23,25,26,29,30,37,38,39,40,43,44,35,36,31,32,27,28,21,22,17,18,14,13,15,16,12,11,20,19,42,41,34,33}}

Answer 2

You could use DepthFirstScan:

Choose the starting vertex:

svertex = Cases[VertexList[g], x_ /; VertexDegree[g, x] == 1]

{9, 33}

Extract the order of vertices:

Reap[DepthFirstScan[g, 
    svertex[[1]], {"PrevisitVertex" -> Sow}];][[2, 1]]

{9, 10, 1, 2, 4, 3, 8, 7, 6, 5, 24, 23, 25, 26, 29, 30, 37, 38, 39, 40, 43, 44, 35, 36, 31, 32, 27, 28, 21, 22, 17, 18, 14, 13, 15, 16, 12, 11, 20, 19, 42, 41, 34, 33}

Answer 3

If g is the graph from the OP, then you can do this in versions 9 through 11

combGrData = {EdgeRules@g, 
    GraphEmbedding@g} /. {{a_Real, b_Real} :> {{a, b}}, 
    HoldPattern@Rule[a__] :> {{a}}};
<< Combinatorica`
Graph @@ combGrData // HamiltonianPath

(* {9, 10, 1, 2, 4, 3, 8, 7, 6, 5, 24, 23, 25, 26, 29, 30, 37, 38, 39, 
    40, 43, 44, 35, 36, 31, 32, 27, 28, 21, 22, 17, 18, 14, 13, 15, 16, 
    12, 11, 20, 19, 42, 41, 34, 33}

If someone can explain the SetDelayed error I get in version 11 I'd be much abliged.