## Exercise ‹26›:

UNDIRECTED HAMILTONIAN CIRCUIT $\leq$ UNDIRECTED HAMILTONIAN PATH
Reduce the UNDIRECTED HAMILTONIAN CIRCUIT problem to the UNDIRECTED HAMILTONIAN PATH problem. These problems are defined as follows:
• UNDIRECTED HAMILTONIAN CIRCUIT: given an undirected graph $G$, is there a cycle that visits each node of $G$ exactly once? In other words, is it possible to order the nodes of $G$ such that there is an edge between each $2$ contiguous nodes and an edge between the last and the first nodes?
More formally, this problem can be defined as the following set:
$\{ G=\langle V,E\rangle \mid V=\{n_1,\ldots,n_k\}\;\wedge\;\forall i\in\{1,\ldots,k\}: \{n_i,n_{(i\text{ mod }k)+1}\}\in E \}$
• UNDIRECTED HAMILTONIAN PATH: given an undirected graph $G$, is there a path that visits each node of $G$ exactly once? In other words, is it possible to order the nodes of $G$ such that there is an edge between each $2$ contiguous nodes?
More formally, this problem can be defined as the following set:
$\{ G=\langle V,E\rangle \mid V=\{n_1,\ldots,n_k\}\;\wedge\;\forall i\in\{1,\ldots,k-1\}: \{n_i,n_{i+1}\}\in E \}$
The input and output of the reduction conform to the following data types:
• in: struct {
numnodes: int
edges: array of array [2] of int
adjacents: array of array of int
adjacencymatrix: array of array of int
}

The input is an undirected graph. The graph is given by means of four different kinds of information: the number of nodes in.numnodes, the list of edges in.edges, for each node the list of its adjacent nodes in.adjacents, and the adjacency matrix in.adjacencymatrix. The nodes are integers between $1$ and in.numnodes, and thus, the $0$’th position of in.adjacents does not contain useful data and should be ignored; for any other position $p$, in.adjacents[p] is the list of nodes adjacent to $p$. For the same reason, the $0$’th row and column of in.adjacencymatrix do not contain useful data and should be ignored; for any other row $r$ and column $c$, in.adjacencymatrix[r][c] is $0$ when there is no edge between the nodes $r$ and $c$, and $1$ otherwise.

Note: The input graph has at least $2$ nodes, no repeated edges, nor self-loop edges.
• out: struct {
edges: array of array [2] of string
nodes: array of string
}

The output is an undirected graph, which is described with the list of edges out.edges and with the list of nodes out.nodes. An edge is a pair of nodes, and each node is represented by either an integer or a string. The list of nodes out.nodes is optional, and contains nodes of the graph, that may or may not appear in the list of edges. This list of nodes could be useful, for example, for including the nodes that are not connected to any other node, if this is considered necessary.

Note: If the output graph has no nodes or just one node, it is considered that it has a path. Repeated edges and self-loop edges are ignored.
Authors: Carles Creus / Documentation:
 main { // Write your reduction here... } To be able to submit you need to either log in, register, or become a guest.