Changes

src/components/EternalDominationGame.tsx

@@ -4,15 +4,18 @@ import { Badge } from "@/components/ui/badge";
 import { Card, CardContent, CardHeader, CardTitle, CardDescription } from "@/components/ui/card";
 import { Shield, Target, RotateCcw, Info } from "lucide-react";
 
-// 12 columns (x: 0-11) × 10 rows (y: 0-9)
+// Hexagonal grid F₃(12, 10) - 12 columns × 10 rows
+// Following the "brick wall" offset pattern for hexagonal grids
 const COLS = 12;
 const ROWS = 10;
 
 type Position = { x: number; y: number };
 
-// Check if a position is on the border of the rectangle
+// Position utilities
-const isBorder = (x: number, y: number): boolean => {
+const posKey = (p: Position): string => `${p.x},${p.y}`;
-  return x === 0 || x === COLS - 1 || y === 0 || y === 1 || y === ROWS - 2 || y === ROWS - 1;
+const parseKey = (key: string): Position => {
+  const [x, y] = key.split(",").map(Number);
+  return { x, y };
 };
 
 // Check if position is valid
@@ -20,35 +23,55 @@ const isValidPos = (x: number, y: number): boolean => {
   return x >= 0 && x < COLS && y >= 0 && y < ROWS;
 };
 
-// Get adjacent positions (4-connectivity for square grid)
+// Hexagonal grid adjacency (brick pattern)
-const getAdjacent = (x: number, y: number): Position[] => {
+// Even rows: neighbors at (-1,0), (1,0), (-1,-1), (0,-1), (-1,1), (0,1)
+// Odd rows: neighbors at (-1,0), (1,0), (0,-1), (1,-1), (0,1), (1,1)
+const getHexNeighbors = (x: number, y: number): Position[] => {
   const neighbors: Position[] = [];
-  const deltas = [[-1, 0], [1, 0], [0, -1], [0, 1]];
+  
-  for (const [dx, dy] of deltas) {
+  // Horizontal neighbors (same for all rows)
-    const nx = x + dx;
+  if (isValidPos(x - 1, y)) neighbors.push({ x: x - 1, y });
-    const ny = y + dy;
+  if (isValidPos(x + 1, y)) neighbors.push({ x: x + 1, y });
-    if (isValidPos(nx, ny)) {
+  
-      neighbors.push({ x: nx, y: ny });
+  // Diagonal neighbors depend on row parity (brick pattern)
-    }
+  const isEvenRow = y % 2 === 0;
+  
+  if (isEvenRow) {
+    // Even row: upper-left, upper-right at x-1 and x
+    if (isValidPos(x - 1, y - 1)) neighbors.push({ x: x - 1, y: y - 1 });
+    if (isValidPos(x, y - 1)) neighbors.push({ x: x, y: y - 1 });
+    // Lower neighbors
+    if (isValidPos(x - 1, y + 1)) neighbors.push({ x: x - 1, y: y + 1 });
+    if (isValidPos(x, y + 1)) neighbors.push({ x: x, y: y + 1 });
+  } else {
+    // Odd row: upper-left, upper-right at x and x+1
+    if (isValidPos(x, y - 1)) neighbors.push({ x: x, y: y - 1 });
+    if (isValidPos(x + 1, y - 1)) neighbors.push({ x: x + 1, y: y - 1 });
+    // Lower neighbors
+    if (isValidPos(x, y + 1)) neighbors.push({ x: x, y: y + 1 });
+    if (isValidPos(x + 1, y + 1)) neighbors.push({ x: x + 1, y: y + 1 });
   }
+  
   return neighbors;
 };
 
-// Position key for Set operations
+// Check if a position is on the border of the "interested rectangle"
-const posKey = (p: Position): string => `${p.x},${p.y}`;
+// Following Figure 10: border forms cycle C around the rectangle
-const parseKey = (key: string): Position => {
+const isBorder = (x: number, y: number): boolean => {
-  const [x, y] = key.split(",").map(Number);
+  return x === 0 || x === COLS - 1 || y === 0 || y === ROWS - 1;
-  return { x, y };
 };
 
-// Initial guard placement based on the paper's configuration
+// Interior positions (not on border)
-// Following the pattern from Theorem 4 (square grid eternal dominating set):
+const isInterior = (x: number, y: number): boolean => {
-// S defined as: (0,0) ∈ S; if (x,y) ∈ S then (x+2,y+1), (x-1,y+2), (x-2,y-1), (x+1,y-2) ∈ S
+  return x > 0 && x < COLS - 1 && y > 0 && y < ROWS - 1;
-// Plus all border vertices are guarded for the finite case
+};
+
+// Generate initial guard placement following the paper's U₃ configuration
+// Interior guards follow a specific pattern, all border vertices are guarded
 const generateInitialGuards = (): Set<string> => {
   const guards = new Set<string>();
   
-  // Add all border vertices (the cycle C from Figure 10)
+  // All border vertices are guarded (forming cycle C)
   for (let x = 0; x < COLS; x++) {
     for (let y = 0; y < ROWS; y++) {
       if (isBorder(x, y)) {
@@ -57,14 +80,15 @@ const generateInitialGuards = (): Set<string> => {
     }
   }
   
-  // Add interior guards following the pattern from Theorem 4
+  // Interior guards following the eternal dominating set pattern
-  // For square grid: (0,0) starts, then (x+2,y+1), (x-1,y+2), etc.
+  // Based on the paper: we need interior guards at density 7/15 nm
-  // We place guards so each interior vertex is dominated by exactly one guard
+  // Pattern: guards at positions creating a dominating set
-  // Pattern: guards at positions where (2x + y) mod 5 === 0
+  // Using pattern similar to Figure 10's interior placement
   for (let x = 1; x < COLS - 1; x++) {
-    for (let y = 2; y < ROWS - 2; y++) {
+    for (let y = 1; y < ROWS - 1; y++) {
-      // Interior region: x in [1, COLS-2], y in [2, ROWS-3]
+      // Place guards following a hexagonal domination pattern
-      // Use the diagonal pattern: place guards at specific intervals
+      // Every vertex in interior needs to be within distance 1 of a guard
+      // Pattern: (2x + y) mod 5 === 0 creates suitable coverage
       if ((2 * x + y) % 5 === 0) {
         guards.add(posKey({ x, y }));
       }
@@ -74,11 +98,11 @@ const generateInitialGuards = (): Set<string> => {
   return guards;
 };
 
-// Check if a vertex is dominated (has a guard on it or adjacent to a guard)
+// Check if a vertex is dominated
 const isDominated = (x: number, y: number, guards: Set<string>): boolean => {
   if (guards.has(posKey({ x, y }))) return true;
-  const adj = getAdjacent(x, y);
+  const neighbors = getHexNeighbors(x, y);
-  return adj.some(p => guards.has(posKey(p)));
+  return neighbors.some(p => guards.has(posKey(p)));
 };
 
 // Check if all vertices are dominated
@@ -91,34 +115,31 @@ const isFullyDominated = (guards: Set<string>): boolean => {
   return true;
 };
 
-// Find the optimal defense move for an attack
+// Defense strategy from Figure 10 and page 23:
-// Returns new guard positions after defending
+// 1. Interior guards shift toward attack
+// 2. If a guard moves from interior to border, border guards shift along complementary paths
 const defendAttack = (guards: Set<string>, attack: Position): Set<string> | null => {
   const attackKey = posKey(attack);
   
-  // If attack is on a guard, no movement needed
+  // If attack is on a guard, already defended
   if (guards.has(attackKey)) {
     return guards;
   }
   
   // Find adjacent guards that can move to defend
-  const adjacent = getAdjacent(attack.x, attack.y);
+  const neighbors = getHexNeighbors(attack.x, attack.y);
-  const adjacentGuards = adjacent.filter(p => guards.has(posKey(p)));
+  const adjacentGuards = neighbors.filter(p => guards.has(posKey(p)));
   
   if (adjacentGuards.length === 0) {
-    return null; // Cannot defend - should not happen with proper dominating set
+    return null; // Should not happen with proper dominating set
   }
   
-  // Strategy based on Figure 10:
-  // 1. Move interior guard toward attack
-  // 2. Shift guards along border paths to maintain coverage
-  
   const newGuards = new Set(guards);
   
-  // Pick the best guard to move (prefer interior guards)
+  // Strategy: prefer interior guards to move (as per Figure 10)
   let defender: Position | null = null;
   for (const g of adjacentGuards) {
-    if (!isBorder(g.x, g.y)) {
+    if (isInterior(g.x, g.y)) {
       defender = g;
       break;
     }
@@ -131,57 +152,71 @@ const defendAttack = (guards: Set<string>, attack: Position): Set<string> | null
   newGuards.delete(posKey(defender));
   newGuards.add(attackKey);
   
-  // If defender was interior and moved to border, we need to shift border guards
+  // If defender was interior and attack is on border, apply border shift strategy
+  // Following the complementary paths P₁,₃ and P₂,₄ from Figure 10
+  if (isInterior(defender.x, defender.y)) {
+    // Interior to border movement: shift guards along border cycle
     // to maintain the dominating set property
-  if (!isBorder(defender.x, defender.y) && isBorder(attack.x, attack.y)) {
+    applyBorderShift(newGuards, defender, attack);
-    // The complementary path strategy from Figure 10
-    // Shift guards along the border cycle to fill the gap left by interior movement
-    performBorderShift(newGuards, attack, defender);
   }
   
-  // Verify we still have a dominating set, if not, apply additional shifts
+  // Ensure domination is maintained
-  ensureDominatingSet(newGuards, attack);
+  ensureDomination(newGuards);
   
   return newGuards;
 };
 
-// Shift guards along border to maintain coverage
+// Border shift strategy: when interior guard moves out, shift border guards
-const performBorderShift = (guards: Set<string>, to: Position, from: Position): void => {
+// along complementary paths to fill gaps
-  // Simple strategy: if we created a gap, try to fill it with neighboring guard shifts
+const applyBorderShift = (guards: Set<string>, from: Position, to: Position): void => {
-  const fromKey = posKey(from);
+  // Check if any interior vertices became undominated
-  
+  for (let x = 1; x < COLS - 1; x++) {
-  // Find border neighbors of the vacated position that have guards
+    for (let y = 1; y < ROWS - 1; y++) {
-  const neighbors = getAdjacent(from.x, from.y);
+      if (!isDominated(x, y, guards)) {
+        // Find a neighboring guard to shift in
+        const neighbors = getHexNeighbors(x, y);
         for (const n of neighbors) {
-    if (isBorder(n.x, n.y) && guards.has(posKey(n))) {
+          if (guards.has(posKey(n))) {
-      // Found a border guard that can help
+            // Check if we can shift this guard
-      // For now, we don't need to shift if the vacated position was interior
+            const nNeighbors = getHexNeighbors(n.x, n.y);
-      break;
+            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 dominating set property after movement
+// Ensure all vertices remain dominated after movement
-const ensureDominatingSet = (guards: Set<string>, attackPos: Position): void => {
+const ensureDomination = (guards: Set<string>): void => {
-  // Check all vertices are still dominated
-  // If not, this would indicate an issue with our defense strategy
-  // In practice, with the border fully guarded, we maintain domination
-  
-  // Add back any critical positions if needed
   for (let x = 0; x < COLS; x++) {
     for (let y = 0; y < ROWS; y++) {
       if (!isDominated(x, y, guards)) {
-        // Find nearest guard and shift toward this position
+        // Emergency: place a guard at nearest available position
-        const adj = getAdjacent(x, y);
+        const neighbors = getHexNeighbors(x, y);
-        for (const a of adj) {
+        for (const n of neighbors) {
-          const adjKey = posKey(a);
+          const nNeighbors = getHexNeighbors(n.x, n.y);
-          // Try to find a guard that can shift
+          for (const nn of nNeighbors) {
-          const adjAdj = getAdjacent(a.x, a.y);
+            if (guards.has(posKey(nn))) {
-          for (const aa of adjAdj) {
-            if (guards.has(posKey(aa)) && !posKey(aa).includes(posKey(attackPos))) {
               // Shift this guard
-              guards.delete(posKey(aa));
+              guards.delete(posKey(nn));
-              guards.add(adjKey);
+              guards.add(posKey(n));
               return;
             }
           }
@@ -194,20 +229,18 @@ const ensureDominatingSet = (guards: Set<string>, attackPos: Position): void =>
 const EternalDominationGame: React.FC = () => {
   const [guards, setGuards] = useState<Set<string>>(generateInitialGuards);
   const [attackHistory, setAttackHistory] = useState<Position[]>([]);
-  const [message, setMessage] = useState<string>("Click any cell to attack. Guards will move to defend.");
+  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 handleCellClick = useCallback((x: number, y: number) => {
     const attack: Position = { x, y };
     
-    // Check if attack is valid (not on a guard)
     if (guards.has(posKey(attack))) {
       setMessage("Cannot attack a guarded position. Choose an unguarded cell.");
       return;
     }
     
-    // Defend the attack
     const newGuards = defendAttack(guards, attack);
     
     if (newGuards) {
@@ -228,65 +261,41 @@ const EternalDominationGame: React.FC = () => {
   const handleReset = useCallback(() => {
     setGuards(generateInitialGuards());
     setAttackHistory([]);
-    setMessage("Click any cell to attack. Guards will move to defend.");
+    setMessage("Click any cell to attack. Guards will shift to defend.");
     setMoveCount(0);
   }, []);
 
-  const getCellStyle = (x: number, y: number): string => {
-    const key = posKey({ x, y });
-    const hasGuard = guards.has(key);
-    const isOnBorder = isBorder(x, y);
-    const dominated = isDominated(x, y, guards);
-    
-    let base = "w-8 h-8 border border-border flex items-center justify-center text-lg cursor-pointer transition-all hover:scale-105 ";
-    
-    if (hasGuard) {
-      base += isOnBorder 
-        ? "bg-primary/30 border-primary" 
-        : "bg-accent/50 border-accent-foreground";
-    } else if (dominated) {
-      base += "bg-muted hover:bg-muted/80";
-    } else {
-      base += "bg-destructive/20 hover:bg-destructive/30"; // Should not happen
-    }
-    
-    return base;
-  };
-
   const guardCount = guards.size;
   const interiorGuards = Array.from(guards).filter(k => {
     const p = parseKey(k);
-    return !isBorder(p.x, p.y);
+    return isInterior(p.x, p.y);
   }).length;
   const borderGuards = guardCount - interiorGuards;
 
+  // Hexagonal cell dimensions
+  const cellWidth = 36;
+  const cellHeight = 32;
+  const rowOffset = cellWidth / 2;
+
   return (
     <div className="space-y-6">
       <Card>
         <CardHeader>
           <CardTitle className="flex items-center gap-2">
             <Shield className="w-5 h-5" />
-            Eternal Domination on a Grid
+            Eternal Domination on a Hexagonal Grid
           </CardTitle>
           <CardDescription>
-            12×10 grid with m-eternal dominating set defense strategy
+            12×10 hexagonal grid F₃(12, 10) with m-eternal dominating set defense
           </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">
+              <Badge variant="outline">Guards: {guardCount}</Badge>
-                Total Guards: {guardCount}
+              <Badge variant="secondary">Border: {borderGuards}</Badge>
-              </Badge>
+              <Badge variant="secondary">Interior: {interiorGuards}</Badge>
-              <Badge variant="secondary">
+              <Badge variant="outline">Attacks: {moveCount}</Badge>
-                Border: {borderGuards}
-              </Badge>
-              <Badge variant="secondary">
-                Interior: {interiorGuards}
-              </Badge>
-              <Badge variant="outline">
-                Attacks: {moveCount}
-              </Badge>
             </div>
             <div className="flex gap-2">
               <Button variant="outline" size="sm" onClick={() => setShowInfo(!showInfo)}>
@@ -302,9 +311,9 @@ const EternalDominationGame: React.FC = () => {
           
           {showInfo && (
             <div className="p-4 bg-muted rounded-lg text-sm space-y-2">
-              <p><strong>m-Eternal Domination:</strong> Guards occupy vertices forming a dominating set. When you attack an unguarded vertex, guards move to defend while maintaining domination of all vertices.</p>
+              <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>Strategy:</strong> Border vertices are always guarded (cycle C). Interior guards shift toward attacks, while border guards shift along complementary paths to maintain coverage.</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>Goal:</strong> The attacker wins if they can find an attack sequence that leaves some vertex undominated. Try to break the defense!</p>
+              <p><strong>Goal:</strong> Try to find an attack sequence that leaves some vertex undominated!</p>
             </div>
           )}
 
@@ -313,53 +322,102 @@ const EternalDominationGame: React.FC = () => {
             {message}
           </div>
 
-          {/* Grid */}
+          {/* Hexagonal Grid */}
           <div className="flex justify-center overflow-x-auto pb-4">
-            <div className="inline-block">
-              <div className="text-xs text-muted-foreground mb-1 pl-4">
-                <div className="flex">
-                  {Array.from({ length: COLS }, (_, i) => (
-                    <div key={i} className="w-8 text-center">{i}</div>
-                  ))}
-                </div>
-              </div>
-              {Array.from({ length: ROWS }, (_, y) => (
-                <div key={y} className="flex items-center">
-                  <div className="w-4 text-xs text-muted-foreground text-right pr-1">
-                    {y}
-                  </div>
-                  {Array.from({ length: COLS }, (_, x) => (
             <div 
-                      key={x}
+              className="relative"
-                      className={getCellStyle(x, y)}
+              style={{ 
-                      onClick={() => handleCellClick(x, y)}
+                width: COLS * cellWidth + rowOffset + 20,
-                      title={`(${x}, ${y}) - ${guards.has(posKey({ x, y })) ? "Guard" : "Empty"}`}
+                height: ROWS * cellHeight + 20
+              }}
             >
-                      {guards.has(posKey({ x, y })) ? "🛡️" : ""}
+              {/* Draw edges first */}
+              <svg 
+                className="absolute inset-0 pointer-events-none"
+                width={COLS * cellWidth + rowOffset + 20}
+                height={ROWS * cellHeight + 20}
+              >
+                {Array.from({ length: ROWS }, (_, y) =>
+                  Array.from({ length: COLS }, (_, x) => {
+                    const offset = y % 2 === 1 ? rowOffset : 0;
+                    const cx = x * cellWidth + cellWidth / 2 + offset + 10;
+                    const cy = y * cellHeight + cellHeight / 2 + 10;
+                    
+                    return getHexNeighbors(x, y)
+                      .filter(n => n.x > x || (n.x === x && n.y > y))
+                      .map((neighbor, i) => {
+                        const nOffset = neighbor.y % 2 === 1 ? rowOffset : 0;
+                        const ncx = neighbor.x * cellWidth + cellWidth / 2 + nOffset + 10;
+                        const ncy = neighbor.y * cellHeight + cellHeight / 2 + 10;
+                        return (
+                          <line
+                            key={`${x}-${y}-${i}`}
+                            x1={cx}
+                            y1={cy}
+                            x2={ncx}
+                            y2={ncy}
+                            stroke="currentColor"
+                            strokeOpacity={0.2}
+                            strokeWidth={1}
+                          />
+                        );
+                      });
+                  })
+                )}
+              </svg>
+
+              {/* Cells */}
+              {Array.from({ length: ROWS }, (_, y) =>
+                Array.from({ length: COLS }, (_, x) => {
+                  const offset = y % 2 === 1 ? rowOffset : 0;
+                  const hasGuard = guards.has(posKey({ x, y }));
+                  const onBorder = isBorder(x, y);
+                  const dominated = isDominated(x, y, guards);
+                  
+                  let bgClass = "bg-muted hover:bg-muted/80";
+                  if (hasGuard) {
+                    bgClass = onBorder 
+                      ? "bg-primary/40 border-primary" 
+                      : "bg-accent border-accent-foreground";
+                  } else if (!dominated) {
+                    bgClass = "bg-destructive/30";
+                  }
+                  
+                  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 ${bgClass}`}
+                      style={{
+                        left: x * cellWidth + offset + 10 + (cellWidth - 28) / 2,
+                        top: y * cellHeight + 10 + (cellHeight - 28) / 2,
+                      }}
+                      onClick={() => handleCellClick(x, y)}
+                      title={`(${x}, ${y}) - ${hasGuard ? "Guard" : dominated ? "Dominated" : "Undominated!"}`}
+                    >
+                      {hasGuard ? "🛡️" : ""}
                     </div>
-                  ))}
+                  );
-                </div>
+                })
-              ))}
+              )}
             </div>
           </div>
 
           {/* 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/30 border border-primary rounded flex items-center justify-center text-xs">🛡️</div>
+              <div className="w-6 h-6 bg-primary/40 border border-primary rounded-full flex items-center justify-center text-xs">🛡️</div>
               <span>Border Guard</span>
             </div>
             <div className="flex items-center gap-2">
-              <div className="w-6 h-6 bg-accent/50 border border-accent-foreground rounded flex items-center justify-center text-xs">🛡️</div>
+              <div className="w-6 h-6 bg-accent border border-accent-foreground 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"></div>
+              <div className="w-6 h-6 bg-muted border border-border rounded-full"></div>
-              <span>Dominated (unguarded)</span>
+              <span>Dominated</span>
             </div>
           </div>
 
-          {/* Attack History */}
           {attackHistory.length > 0 && (
             <div className="text-sm">
               <p className="text-muted-foreground mb-1">Recent attacks:</p>