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    <p>The best way to do this I think is by calculating all valid paths of length at most <em>k</em> between the source <em>S</em> and every other node, then using that information to calculate all valid paths of length at most <em>k+1</em>. You then just repeat this until you get a fixed point where no paths are modified.</p> <p>In practice, this means you should set up a list of paths at each node. At each step, you take each node <em>U</em> in turn and look at the paths that, in the previous step, terminated at some neighbour <em>V</em> of <em>U</em>. If any of those paths can be extended to be a new, distinct path to <em>U</em>, extend it and add it to <em>U</em>'s list.</p> <p>If when you execute a step you find no new paths, that's your termination state. You can then check the list of paths at the target node <em>T</em>.</p> <p>Pseudocode (in a very loose C# formalism):</p> <pre><code>var paths = graph.nodes.ToDictionary(node =&gt; node, node =&gt; new List&lt;List&lt;node&gt;&gt;()) paths[S].Add(new List&lt;node&gt; {S}) // The trivial path that'll start us off. bool notAFixedPoint = true; while (notAFixedPoint) { notAFixedPoint = false // Assume we're not gonna find any new paths. foreach (var node in graph) { var pathsToNode = paths[node] foreach (var neighbour in node.Neighbours) { var pathsToNeighbour = paths[neighbour] // ExtendPaths is where all the logic about how to recognise a valid path goes. var newPathsToNode = ExtendPaths(pathsToNeighbour, node) // The use of "Except" here is for expository purposes. It wouldn't actually work, // because collections in most languages are compared by reference rather than by value. if (newPathsToNode.Except(pathsToNode).IsNotEmpty()) { // We've found some new paths, so we can't terminate yet. notAFixedPoint = true pathsToNode.AddMany(newPathsToNode) } } } } return paths[T] </code></pre>
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    1. COSo, for the first portion, I would make paths from source S to its neighbors, and have a length of k. Then, at length k+1, this will add paths from nodes that are adjacent to the nodes connected to the source, and in that way give me the paths from source up to those nodes. And iterate this until I get the target node in the result? How can I know how to stop, since multiple paths might be possible. Could you guide me a bit with some pseudo code, I really don't have a lot of experience with writing this kind of stuff. Thanks!
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    2. CODoes the edit make more sense?
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    3. COThanks for the input and your insight, I spent some more time with this and arrived at solution from a different angle, the other portion besides this pseudocode of yours are the constrains, which were "the key" of this issue.
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