Changes

src/components/EternalDominationGame.tsx

@@ -55,23 +55,54 @@ const getHexNeighbors = (x: number, y: number): Position[] => {
   return neighbors;
 };
 
-// Check if a position is on the border of the "interested rectangle"
+// Following Figure 10: the border is a DOUBLE LAYER forming cycle C
-// Following Figure 10: border forms cycle C around the rectangle
+// Outer border: x=0, x=COLS-1, y=0, y=ROWS-1
+// Inner border: x=1, x=COLS-2, y=1, y=ROWS-2 (but only the perimeter)
 const isBorder = (x: number, y: number): boolean => {
-  return x === 0 || x === COLS - 1 || y === 0 || y === ROWS - 1;
+  // Outer layer
+  if (x === 0 || x === COLS - 1 || y === 0 || y === ROWS - 1) return true;
+  // Inner layer (second ring)
+  if (x === 1 || x === COLS - 2 || y === 1 || y === ROWS - 2) return true;
+  return false;
 };
 
-// Interior positions (not on border)
+// Interior positions (inside the double border)
 const isInterior = (x: number, y: number): boolean => {
-  return x > 0 && x < COLS - 1 && y > 0 && y < ROWS - 1;
+  return x > 1 && x < COLS - 2 && y > 1 && y < ROWS - 2;
+};
+
+// Get the border layer (0 = outer, 1 = inner)
+const getBorderLayer = (x: number, y: number): number => {
+  if (x === 0 || x === COLS - 1 || y === 0 || y === ROWS - 1) return 0;
+  if (x === 1 || x === COLS - 2 || y === 1 || y === ROWS - 2) return 1;
+  return -1;
+};
+
+// Get all border vertices in order (forming cycle C)
+const getBorderCycle = (): Position[] => {
+  const border: Position[] = [];
+  
+  // Outer ring clockwise: top -> right -> bottom -> left
+  for (let x = 0; x < COLS; x++) border.push({ x, y: 0 });
+  for (let y = 1; y < ROWS; y++) border.push({ x: COLS - 1, y });
+  for (let x = COLS - 2; x >= 0; x--) border.push({ x, y: ROWS - 1 });
+  for (let y = ROWS - 2; y >= 1; y--) border.push({ x: 0, y });
+  
+  // Inner ring clockwise
+  for (let x = 1; x < COLS - 1; x++) border.push({ x, y: 1 });
+  for (let y = 2; y < ROWS - 1; y++) border.push({ x: COLS - 2, y });
+  for (let x = COLS - 3; x >= 1; x--) border.push({ x, y: ROWS - 2 });
+  for (let y = ROWS - 3; y >= 2; y--) border.push({ x: 1, y });
+  
+  return border;
 };
 
 // Generate initial guard placement following the paper's U₃ configuration
-// Interior guards follow a specific pattern, all border vertices are guarded
+// ALL border vertices are guarded (double layer), plus interior guards in dominating pattern
 const generateInitialGuards = (): Set<string> => {
   const guards = new Set<string>();
   
-  // All border vertices are guarded (forming cycle C)
+  // All border vertices are guarded (both layers - forming double protection)
   for (let x = 0; x < COLS; x++) {
     for (let y = 0; y < ROWS; y++) {
       if (isBorder(x, y)) {
@@ -80,16 +111,14 @@ const generateInitialGuards = (): Set<string> => {
     }
   }
   
-  // Interior guards following the eternal dominating set pattern
+  // Interior guards following the U₃ eternal dominating set pattern
-  // Based on the paper: we need interior guards at density 7/15 nm
+  // Interior is 8x6 grid (from x=2 to x=9, y=2 to y=7)
-  // Pattern: guards at positions creating a dominating set
+  // Place guards to ensure domination at density ~7/15
-  // Using pattern similar to Figure 10's interior placement
+  for (let x = 2; x < COLS - 2; x++) {
-  for (let x = 1; x < COLS - 1; x++) {
+    for (let y = 2; y < ROWS - 2; y++) {
-    for (let y = 1; y < ROWS - 1; y++) {
+      // Pattern based on Figure 10 interior placement
-      // Place guards following a hexagonal domination pattern
+      // Creates a dominating set where every interior vertex is within distance 1 of a guard
-      // Every vertex in interior needs to be within distance 1 of a guard
+      if ((x + 2 * y) % 4 === 0) {
-      // Pattern: (2x + y) mod 5 === 0 creates suitable coverage
-      if ((2 * x + y) % 5 === 0) {
         guards.add(posKey({ x, y }));
       }
     }
@@ -115,15 +144,41 @@ const isFullyDominated = (guards: Set<string>): boolean => {
   return true;
 };
 
+// Find a path along the border cycle between two positions
+const findBorderPath = (from: Position, to: Position, borderCycle: Position[]): Position[] => {
+  const fromIdx = borderCycle.findIndex(p => p.x === from.x && p.y === from.y);
+  const toIdx = borderCycle.findIndex(p => p.x === to.x && p.y === to.y);
+  
+  if (fromIdx === -1 || toIdx === -1) return [];
+  
+  const path: Position[] = [];
+  const n = borderCycle.length;
+  
+  // Go clockwise
+  let i = fromIdx;
+  while (i !== toIdx) {
+    path.push(borderCycle[i]);
+    i = (i + 1) % n;
+  }
+  path.push(borderCycle[toIdx]);
+  
+  return path;
+};
+
 // Defense strategy from Figure 10 and page 23:
-// 1. Interior guards shift toward attack
+// 1. Interior guards shift toward attack (one guard moves to v_t)
-// 2. If a guard moves from interior to border, border guards shift along complementary paths
+// 2. This may push guards from interior to border (u₁→w₁, u₂→w₂)
-const defendAttack = (guards: Set<string>, attack: Position): Set<string> | null => {
+// 3. Guards shift from border to interior to fill gaps (w₃→u₃, w₄→u₄)
+// 4. Border guards shift along complementary paths P₁,₃ and P₂,₄ to restore configuration
+const defendAttack = (guards: Set<string>, attack: Position): { 
+  newGuards: Set<string>; 
+  movements: Array<{ from: Position; to: Position; type: 'interior' | 'border' | 'path' }>
+} | null => {
   const attackKey = posKey(attack);
   
   // If attack is on a guard, already defended
   if (guards.has(attackKey)) {
-    return guards;
+    return { newGuards: guards, movements: [] };
   }
   
   // Find adjacent guards that can move to defend
@@ -131,12 +186,15 @@ const defendAttack = (guards: Set<string>, attack: Position): Set<string> | null
   const adjacentGuards = neighbors.filter(p => guards.has(posKey(p)));
   
   if (adjacentGuards.length === 0) {
-    return null; // Should not happen with proper dominating set
+    return null;
   }
   
   const newGuards = new Set(guards);
+  const movements: Array<{ from: Position; to: Position; type: 'interior' | 'border' | 'path' }> = [];
+  const borderCycle = getBorderCycle();
   
-  // Strategy: prefer interior guards to move (as per Figure 10)
+  // Strategy: prefer interior guards to move to attack position (as per Figure 10)
+  // This simulates the "internal shift toward attack"
   let defender: Position | null = null;
   for (const g of adjacentGuards) {
     if (isInterior(g.x, g.y)) {
@@ -144,6 +202,16 @@ const defendAttack = (guards: Set<string>, attack: Position): Set<string> | null
       break;
     }
   }
+  
+  // If no interior guard adjacent, use inner border guard, then outer border
+  if (!defender) {
+    for (const g of adjacentGuards) {
+      if (getBorderLayer(g.x, g.y) === 1) {
+        defender = g;
+        break;
+      }
+    }
+  }
   if (!defender) {
     defender = adjacentGuards[0];
   }
@@ -151,79 +219,77 @@ const defendAttack = (guards: Set<string>, attack: Position): Set<string> | null
   // Move defender to attack position
   newGuards.delete(posKey(defender));
   newGuards.add(attackKey);
+  movements.push({ from: defender, to: attack, type: 'interior' });
   
-  // If defender was interior and attack is on border, apply border shift strategy
+  // Now we need to restore the configuration:
-  // Following the complementary paths P₁,₃ and P₂,₄ from Figure 10
+  // If an interior guard moved, we need to shift guards along the border to fill the gap
-  if (isInterior(defender.x, defender.y)) {
+  // Following Figure 10: find complementary paths to shift guards
-    // Interior to border movement: shift guards along border cycle
-    // to maintain the dominating set property
-    applyBorderShift(newGuards, defender, attack);
-  }
   
-  // Ensure domination is maintained
+  if (isInterior(defender.x, defender.y) || getBorderLayer(defender.x, defender.y) === 1) {
-  ensureDomination(newGuards);
+    // Find undominated interior vertices and fill from border
+    for (let x = 2; x < COLS - 2; x++) {
+      for (let y = 2; y < ROWS - 2; y++) {
+        if (!isDominated(x, y, newGuards)) {
+          // Find adjacent border guard to move in
+          const nbs = getHexNeighbors(x, y);
+          const borderNb = nbs.find(n => 
+            isBorder(n.x, n.y) && newGuards.has(posKey(n))
+          );
           
-  return newGuards;
+          if (borderNb) {
-};
+            // Move border guard to interior (w₃→u₃ or w₄→u₄)
+            newGuards.delete(posKey(borderNb));
+            newGuards.add(posKey({ x, y }));
+            movements.push({ from: borderNb, to: { x, y }, type: 'border' });
             
-// Border shift strategy: when interior guard moves out, shift border guards
+            // Now shift guards along border path to fill the gap at borderNb
-// along complementary paths to fill gaps
+            // Find another border guard to shift toward borderNb
-const applyBorderShift = (guards: Set<string>, from: Position, to: Position): void => {
+            const borderNbNeighbors = getHexNeighbors(borderNb.x, borderNb.y);
-  // Check if any interior vertices became undominated
+            const borderShifter = borderNbNeighbors.find(n =>
-  for (let x = 1; x < COLS - 1; x++) {
+              isBorder(n.x, n.y) && newGuards.has(posKey(n)) && 
-    for (let y = 1; y < ROWS - 1; y++) {
+              !movements.some(m => m.from.x === n.x && m.from.y === n.y)
-      if (!isDominated(x, y, guards)) {
-        // Find a neighboring guard to shift in
-        const neighbors = getHexNeighbors(x, y);
-        for (const n of neighbors) {
-          if (guards.has(posKey(n))) {
-            // Check if we can shift this guard
-            const nNeighbors = getHexNeighbors(n.x, n.y);
-            const canShift = nNeighbors.some(nn => 
-              guards.has(posKey(nn)) && 
-              posKey(nn) !== posKey(n) &&
-              posKey(nn) !== posKey(to)
             );
-            if (canShift || isBorder(n.x, n.y)) {
-              // Shift a border guard toward the gap
-              const borderNeighbors = neighbors.filter(nb => 
-                isBorder(nb.x, nb.y) && guards.has(posKey(nb))
-              );
-              if (borderNeighbors.length > 0) {
-                const shifter = borderNeighbors[0];
-                guards.delete(posKey(shifter));
-                guards.add(posKey({ x, y }));
-                return;
-              }
-            }
-          }
-        }
-      }
-    }
-  }
-};
             
-// Ensure all vertices remain dominated after movement
+            if (borderShifter) {
-const ensureDomination = (guards: Set<string>): void => {
+              // Shift along complementary path
-  for (let x = 0; x < COLS; x++) {
+              newGuards.delete(posKey(borderShifter));
-    for (let y = 0; y < ROWS; y++) {
+              newGuards.add(posKey(borderNb));
-      if (!isDominated(x, y, guards)) {
+              movements.push({ from: borderShifter, to: borderNb, type: 'path' });
-        // Emergency: place a guard at nearest available position
-        const neighbors = getHexNeighbors(x, y);
-        for (const n of neighbors) {
-          const nNeighbors = getHexNeighbors(n.x, n.y);
-          for (const nn of nNeighbors) {
-            if (guards.has(posKey(nn))) {
-              // Shift this guard
-              guards.delete(posKey(nn));
-              guards.add(posKey(n));
-              return;
             }
           }
         }
       }
     }
+    
+    // Ensure border remains fully covered by shifting along paths
+    for (const pos of borderCycle) {
+      if (!newGuards.has(posKey(pos))) {
+        // Find adjacent border guard to shift
+        const nbs = getHexNeighbors(pos.x, pos.y);
+        const shifter = nbs.find(n => 
+          isBorder(n.x, n.y) && newGuards.has(posKey(n)) &&
+          !movements.some(m => m.to.x === n.x && m.to.y === n.y && m.type === 'path')
+        );
+        
+        if (shifter) {
+          // Check if shifter has another border neighbor that can cover its old spot
+          const shifterNbs = getHexNeighbors(shifter.x, shifter.y);
+          const backfill = shifterNbs.find(n =>
+            isBorder(n.x, n.y) && newGuards.has(posKey(n)) && 
+            (n.x !== pos.x || n.y !== pos.y)
+          );
+          
+          if (backfill) {
+            newGuards.delete(posKey(shifter));
+            newGuards.add(posKey(pos));
+            movements.push({ from: shifter, to: pos, type: 'path' });
+          }
+        }
+      }
+    }
   }
+  
+  return { newGuards, movements };
 };
 
 const EternalDominationGame: React.FC = () => {
@@ -232,6 +298,7 @@ const EternalDominationGame: React.FC = () => {
   const [message, setMessage] = useState<string>("Click any cell to attack. Guards will shift to defend.");
   const [moveCount, setMoveCount] = useState(0);
   const [showInfo, setShowInfo] = useState(false);
+  const [lastMovements, setLastMovements] = useState<Array<{ from: Position; to: Position; type: string }>>([]);
 
   const handleCellClick = useCallback((x: number, y: number) => {
     const attack: Position = { x, y };
@@ -241,15 +308,19 @@ const EternalDominationGame: React.FC = () => {
       return;
     }
     
-    const newGuards = defendAttack(guards, attack);
+    const result = defendAttack(guards, attack);
     
-    if (newGuards) {
+    if (result) {
-      setGuards(newGuards);
+      setGuards(result.newGuards);
+      setLastMovements(result.movements);
       setAttackHistory(prev => [...prev, attack]);
       setMoveCount(prev => prev + 1);
       
-      if (isFullyDominated(newGuards)) {
+      if (isFullyDominated(result.newGuards)) {
-        setMessage(`Attack defended! All vertices remain dominated. (Move ${moveCount + 1})`);
+        const moveDesc = result.movements.length > 1 
+          ? `${result.movements.length} guards shifted` 
+          : "1 guard moved";
+        setMessage(`Attack defended! ${moveDesc}. All vertices remain dominated. (Move ${moveCount + 1})`);
       } else {
         setMessage("Defense failed! Some vertices are not dominated.");
       }
@@ -261,6 +332,7 @@ const EternalDominationGame: React.FC = () => {
   const handleReset = useCallback(() => {
     setGuards(generateInitialGuards());
     setAttackHistory([]);
+    setLastMovements([]);
     setMessage("Click any cell to attack. Guards will shift to defend.");
     setMoveCount(0);
   }, []);
@@ -273,8 +345,8 @@ const EternalDominationGame: React.FC = () => {
   const borderGuards = guardCount - interiorGuards;
 
   // Hexagonal cell dimensions
-  const cellWidth = 36;
+  const cellWidth = 40;
-  const cellHeight = 32;
+  const cellHeight = 36;
   const rowOffset = cellWidth / 2;
 
   return (
@@ -286,15 +358,15 @@ const EternalDominationGame: React.FC = () => {
             Eternal Domination on a Hexagonal Grid
           </CardTitle>
           <CardDescription>
-            12×10 hexagonal grid F₃(12, 10) with m-eternal dominating set defense
+            12×10 hexagonal grid with double-layer border defense (Figure 10)
           </CardDescription>
         </CardHeader>
         <CardContent className="space-y-4">
           <div className="flex flex-wrap gap-2 items-center justify-between">
             <div className="flex gap-2">
               <Badge variant="outline">Guards: {guardCount}</Badge>
-              <Badge variant="secondary">Border: {borderGuards}</Badge>
+              <Badge variant="secondary" className="bg-cyan-500/20 text-cyan-700 dark:text-cyan-300">Border: {borderGuards}</Badge>
-              <Badge variant="secondary">Interior: {interiorGuards}</Badge>
+              <Badge variant="secondary" className="bg-red-500/20 text-red-700 dark:text-red-300">Interior: {interiorGuards}</Badge>
               <Badge variant="outline">Attacks: {moveCount}</Badge>
             </div>
             <div className="flex gap-2">
@@ -311,8 +383,8 @@ const EternalDominationGame: React.FC = () => {
           
           {showInfo && (
             <div className="p-4 bg-muted rounded-lg text-sm space-y-2">
-              <p><strong>m-Eternal Domination on Hexagonal Grid:</strong> Guards form a dominating set on F₃(n,m). Each attack must be defended by moving guards while maintaining domination.</p>
+              <p><strong>m-Eternal Domination (Figure 10):</strong> The border forms a double-layer cycle C with guards on every vertex. Interior guards follow U₃ configuration.</p>
-              <p><strong>Strategy (Figure 10):</strong> Border vertices form cycle C (always guarded). Interior guards shift toward attacks. Complementary paths along the border shift guards to fill gaps.</p>
+              <p><strong>Defense Strategy:</strong> When attacked, interior guards shift toward attack. Guards may move from interior→border or border→interior. Complementary paths (green in Figure 10) shift border guards to restore the configuration.</p>
               <p><strong>Goal:</strong> Try to find an attack sequence that leaves some vertex undominated!</p>
             </div>
           )}
@@ -331,7 +403,7 @@ const EternalDominationGame: React.FC = () => {
                 height: ROWS * cellHeight + 20
               }}
             >
-              {/* Draw edges first */}
+              {/* Draw edges */}
               <svg 
                 className="absolute inset-0 pointer-events-none"
                 width={COLS * cellWidth + rowOffset + 20}
@@ -349,6 +421,10 @@ const EternalDominationGame: React.FC = () => {
                         const nOffset = neighbor.y % 2 === 1 ? rowOffset : 0;
                         const ncx = neighbor.x * cellWidth + cellWidth / 2 + nOffset + 10;
                         const ncy = neighbor.y * cellHeight + cellHeight / 2 + 10;
+                        
+                        // Color border edges green (complementary paths)
+                        const isBorderEdge = isBorder(x, y) && isBorder(neighbor.x, neighbor.y);
+                        
                         return (
                           <line
                             key={`${x}-${y}-${i}`}
@@ -356,14 +432,54 @@ const EternalDominationGame: React.FC = () => {
                             y1={cy}
                             x2={ncx}
                             y2={ncy}
-                            stroke="currentColor"
+                            stroke={isBorderEdge ? "#22c55e" : "currentColor"}
-                            strokeOpacity={0.2}
+                            strokeOpacity={isBorderEdge ? 0.6 : 0.2}
-                            strokeWidth={1}
+                            strokeWidth={isBorderEdge ? 2 : 1}
                           />
                         );
                       });
                   })
                 )}
+                
+                {/* Draw movement arrows for last defense */}
+                {lastMovements.map((move, i) => {
+                  const fromOffset = move.from.y % 2 === 1 ? rowOffset : 0;
+                  const toOffset = move.to.y % 2 === 1 ? rowOffset : 0;
+                  const x1 = move.from.x * cellWidth + cellWidth / 2 + fromOffset + 10;
+                  const y1 = move.from.y * cellHeight + cellHeight / 2 + 10;
+                  const x2 = move.to.x * cellWidth + cellWidth / 2 + toOffset + 10;
+                  const y2 = move.to.y * cellHeight + cellHeight / 2 + 10;
+                  
+                  const color = move.type === 'path' ? '#22c55e' : move.type === 'border' ? '#06b6d4' : '#ef4444';
+                  
+                  return (
+                    <g key={`move-${i}`}>
+                      <line
+                        x1={x1}
+                        y1={y1}
+                        x2={x2}
+                        y2={y2}
+                        stroke={color}
+                        strokeWidth={3}
+                        strokeOpacity={0.8}
+                        markerEnd="url(#arrowhead)"
+                      />
+                    </g>
+                  );
+                })}
+                
+                <defs>
+                  <marker
+                    id="arrowhead"
+                    markerWidth="10"
+                    markerHeight="7"
+                    refX="9"
+                    refY="3.5"
+                    orient="auto"
+                  >
+                    <polygon points="0 0, 10 3.5, 0 7" fill="#ef4444" />
+                  </marker>
+                </defs>
               </svg>
 
               {/* Cells */}
@@ -372,27 +488,34 @@ const EternalDominationGame: React.FC = () => {
                   const offset = y % 2 === 1 ? rowOffset : 0;
                   const hasGuard = guards.has(posKey({ x, y }));
                   const onBorder = isBorder(x, y);
+                  const borderLayer = getBorderLayer(x, y);
                   const dominated = isDominated(x, y, guards);
                   
                   let bgClass = "bg-muted hover:bg-muted/80";
                   if (hasGuard) {
-                    bgClass = onBorder 
+                    if (onBorder) {
-                      ? "bg-primary/40 border-primary" 
+                      // Cyan for border guards (both layers)
-                      : "bg-accent border-accent-foreground";
+                      bgClass = borderLayer === 0 
+                        ? "bg-cyan-500/50 border-cyan-600" 
+                        : "bg-cyan-400/40 border-cyan-500";
+                    } else {
+                      // Red/accent for interior guards
+                      bgClass = "bg-red-500/50 border-red-600";
+                    }
                   } else if (!dominated) {
-                    bgClass = "bg-destructive/30";
+                    bgClass = "bg-destructive/50 border-destructive";
                   }
                   
                   return (
                     <div
                       key={`${x}-${y}`}
-                      className={`absolute w-7 h-7 rounded-full border flex items-center justify-center text-sm cursor-pointer transition-all hover:scale-110 ${bgClass}`}
+                      className={`absolute w-8 h-8 rounded-full border-2 flex items-center justify-center text-sm cursor-pointer transition-all hover:scale-110 ${bgClass}`}
                       style={{
-                        left: x * cellWidth + offset + 10 + (cellWidth - 28) / 2,
+                        left: x * cellWidth + offset + 10 + (cellWidth - 32) / 2,
-                        top: y * cellHeight + 10 + (cellHeight - 28) / 2,
+                        top: y * cellHeight + 10 + (cellHeight - 32) / 2,
                       }}
                       onClick={() => handleCellClick(x, y)}
-                      title={`(${x}, ${y}) - ${hasGuard ? "Guard" : dominated ? "Dominated" : "Undominated!"}`}
+                      title={`(${x}, ${y}) - ${hasGuard ? (onBorder ? "Border Guard" : "Interior Guard") : dominated ? "Dominated" : "Undominated!"}`}
                     >
                       {hasGuard ? "🛡️" : ""}
                     </div>
@@ -405,16 +528,16 @@ const EternalDominationGame: React.FC = () => {
           {/* Legend */}
           <div className="flex flex-wrap gap-4 text-sm">
             <div className="flex items-center gap-2">
-              <div className="w-6 h-6 bg-primary/40 border border-primary rounded-full flex items-center justify-center text-xs">🛡️</div>
+              <div className="w-6 h-6 bg-cyan-500/50 border-2 border-cyan-600 rounded-full flex items-center justify-center text-xs">🛡️</div>
-              <span>Border Guard</span>
+              <span>Border Guard (double layer)</span>
             </div>
             <div className="flex items-center gap-2">
-              <div className="w-6 h-6 bg-accent border border-accent-foreground rounded-full flex items-center justify-center text-xs">🛡️</div>
+              <div className="w-6 h-6 bg-red-500/50 border-2 border-red-600 rounded-full flex items-center justify-center text-xs">🛡️</div>
               <span>Interior Guard</span>
             </div>
             <div className="flex items-center gap-2">
-              <div className="w-6 h-6 bg-muted border border-border rounded-full"></div>
+              <div className="w-3 h-0.5 bg-green-500"></div>
-              <span>Dominated</span>
+              <span>Complementary paths (border cycle)</span>
             </div>
           </div>