{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Dominating set.\n",
    "\n",
    "**The problem**: Given an undirected graph,\n",
    "the dominating set problem is to select a subset of vertices such that each remaining vertex is adjacent\n",
    "to some selected vertex, where the number of selected vertices must not exceed a given threshold.\n",
    "\n",
    "**An example:**\n",
    "The graph $(\\{1,2,3,4,5,6\\},\\{\\{1,2\\},\\{1,3\\},\\{2,4\\},\\{3,5\\},\\{4,5\\},\\{4,6\\}\\})$\n",
    "has two dominating sets with at most $2$ vertices: $\\{1,4\\}$ and $\\{3,4\\}$.\n",
    "\n",
    "**Instance format**: We represent such a problem instance by facts as follows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "%%file instances/instance-dominating.lp\n",
    "vertex(1). vertex(2). vertex(3). vertex(4). vertex(5). vertex(6).\n",
    "edge(1,2). edge(1,3). edge(2,4). edge(3,5). edge(4,5). edge(4,6).\n",
    "threshold(2)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**The task**: Specify a uniform problem encoding such that atoms over the predicate ``select/1`` within the stable models correspond to dominating sets for arbitrary instances (see the output under **Help** below)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%clingo 0 instances/instance-dominating.lp -\n",
    "\n",
    "% your encoding goes here...\n",
    "\n",
    "% show\n",
    "#show select/1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The symbol ``-`` in the command call tells ``clingo`` to read the instance file _and_ the content of the cell. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Help**: The next cell contains a hard to read encoding for the problem that\n",
    "you can use for testing your solution.\n",
    "A readable version of it can be found in the ``solutions`` directory."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "clingo version 5.4.0\n",
      "Reading from instances/instance-dominating.lp ...\n",
      "Solving...\n",
      "Answer: 1\n",
      "select(1) select(4)\n",
      "Answer: 2\n",
      "select(3) select(4)\n",
      "SATISFIABLE\n",
      "\n",
      "Models       : 2\n",
      "Calls        : 1\n",
      "Time         : 0.002s (Solving: 0.00s 1st Model: 0.00s Unsat: 0.00s)\n",
      "CPU Time     : 0.002s\n"
     ]
    }
   ],
   "source": [
    "%%clingo 0 instances/instance-dominating.lp -\n",
    "#show select/1.ssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43):-select(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43).sodjhoinfeorfioejoi2423onjnfeiooi454onro3jfo34oi43o43nfo34oi34iofj43(R0JER434343rfeferfergerGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43):-threshold(R0JER434343rfeferfergerGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43).:-vertex(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43),not ssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43).ssdfefcerverversodjhoinfeorfioejoi2423onjnfeiooi454onro3jfo34oi43o43nfo34oi34iofj43(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43,RJER434343rfeferfergerGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43):-edge(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43,RJER434343rfeferfergerGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43).scerverversodjhoinfeorfioejoi2423onjnfeiooi454onro3jfo34oi43o43nfo34oi34iofj43(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43):-vertex(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43).ssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43(RJER434343rfeferfergerGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43):-select(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43),edge(RJER434343rfeferfergerGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43,RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43).ssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43(RJER434343rfeferfergerGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43):-select(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43),edge(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43,RJER434343rfeferfergerGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43).select(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43):-ssdfefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43).{ssdfefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43):vertex(RJGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43)}R0JER434343rfeferfergerGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43:-threshold(R0JER434343rfeferfergerGRORGssdergergergregrefefcerverversodjhoinfeorfioejoi2423onjnfeiegegegegdcsvooi454onro3jfo34oi43o43nfo34oi34iofj43)."
   ]
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