I stared at the examples for longer than I should have before I realized this is just an Eulerian path problem on a 26-node directed graph where each word is an edge from its first letter to its last.
Model the problem as finding an Eulerian trail in a directed multigraph where each word is an edge from its first to last character. Use Hierholzer's algorithm to find the trail in O(V+E) time, then reconstruct the merged string by overlapping characters. Address edge cases like self-loops, disconnected components, and degree conditions explicitly.
Pro tip: Emphasize that the graph has at most 26 vertices (letters), so the algorithm is effectively linear in the number of words. Also, mention that you would validate the degree conditions before running Hierholzer to quickly return 'IMPOSSIBLE'.
Represent each word as a directed edge from its first character to its last character. Note that multiple edges (words) between the same pair of vertices are allowed, forming a multigraph.
Check that the graph is weakly connected (ignoring isolated vertices) and that the degree conditions for an Eulerian trail are met: at most one vertex with out-degree - in-degree = 1 (start), at most one with in-degree - out-degree = 1 (end), and all others balanced.
If conditions hold, run Hierholzer's algorithm to find an Eulerian trail. Use a stack to perform a depth-first traversal, splicing cycles together, ensuring O(V+E) time.
Traverse the Eulerian trail in order, appending each word's characters except the first (since it overlaps with the previous word's last character). Handle the first word specially by appending all its characters.
Explicitly discuss self-loops (words with same first and last character) which contribute equally to in and out degrees and are naturally handled. For disconnected components, ensure all edges belong to a single weakly connected component; otherwise return 'IMPOSSIBLE'.
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