diff --git a/src/game/chase/hexGraph.ts b/src/game/chase/hexGraph.ts
new file mode 100644
index 0000000..6361d78
--- /dev/null
+++ b/src/game/chase/hexGraph.ts
@@ -0,0 +1,71 @@
+import type { GameGraph, NodeId } from "./types";
+
+export function axialNeighbors(q: number, r: number): Array<[number, number]> {
+  return [
+    [q + 1, r],
+    [q + 1, r - 1],
+    [q, r - 1],
+    [q - 1, r],
+    [q - 1, r + 1],
+    [q, r + 1],
+  ];
+}
+
+export function shortestPathLength(
+  graph: GameGraph,
+  from: NodeId,
+  to: NodeId,
+): number {
+  if (from === to) {
+    return 0;
+  }
+
+  if (!graph.nodes[from] || !graph.nodes[to]) {
+    return Infinity;
+  }
+
+  const queue: Array<[NodeId, number]> = [[from, 0]];
+  const visited = new Set<NodeId>([from]);
+
+  while (queue.length > 0) {
+    const [current, distance] = queue.shift()!;
+    const node = graph.nodes[current];
+
+    for (const neighbor of node.neighbors) {
+      if (neighbor === to) {
+        return distance + 1;
+      }
+      if (!visited.has(neighbor) && graph.nodes[neighbor]) {
+        visited.add(neighbor);
+        queue.push([neighbor, distance + 1]);
+      }
+    }
+  }
+
+  return Infinity;
+}
+
+export function isConnected(graph: GameGraph): boolean {
+  const nodeIds = Object.keys(graph.nodes);
+  if (nodeIds.length <= 1) {
+    return true;
+  }
+
+  const start = nodeIds[0];
+  const visited = new Set<NodeId>([start]);
+  const queue: NodeId[] = [start];
+
+  while (queue.length > 0) {
+    const current = queue.shift()!;
+    const node = graph.nodes[current];
+
+    for (const neighbor of node.neighbors) {
+      if (!visited.has(neighbor) && graph.nodes[neighbor]) {
+        visited.add(neighbor);
+        queue.push(neighbor);
+      }
+    }
+  }
+
+  return visited.size === nodeIds.length;
+}
diff --git a/tests/chase/hexGraph.test.ts b/tests/chase/hexGraph.test.ts
new file mode 100644
index 0000000..d8fd6de
--- /dev/null
+++ b/tests/chase/hexGraph.test.ts
@@ -0,0 +1,67 @@
+import { describe, expect, it } from "vitest";
+import {
+  axialNeighbors,
+  isConnected,
+  shortestPathLength,
+} from "../../src/game/chase/hexGraph";
+import type { GameGraph } from "../../src/game/chase/types";
+
+function buildGraph(nodes: GameGraph["nodes"]): GameGraph {
+  return {
+    nodes,
+    edgeList: [],
+  };
+}
+
+describe("hex graph utilities", () => {
+  it("returns 6 axial neighbors", () => {
+    const neighbors = axialNeighbors(0, 0);
+    expect(neighbors).toEqual([
+      [1, 0],
+      [1, -1],
+      [0, -1],
+      [-1, 0],
+      [-1, 1],
+      [0, 1],
+    ]);
+  });
+
+  it("finds bfs shortest path length", () => {
+    const graph = buildGraph({
+      A: { id: "A", q: 0, r: 0, neighbors: ["B"] },
+      B: { id: "B", q: 1, r: 0, neighbors: ["A", "C"] },
+      C: { id: "C", q: 2, r: 0, neighbors: ["B", "D"] },
+      D: { id: "D", q: 3, r: 0, neighbors: ["C"] },
+    });
+
+    expect(shortestPathLength(graph, "A", "D")).toBe(3);
+    expect(shortestPathLength(graph, "A", "A")).toBe(0);
+  });
+
+  it("returns Infinity when target is unreachable", () => {
+    const graph = buildGraph({
+      A: { id: "A", q: 0, r: 0, neighbors: ["B"] },
+      B: { id: "B", q: 1, r: 0, neighbors: ["A"] },
+      C: { id: "C", q: 5, r: 5, neighbors: [] },
+    });
+
+    expect(shortestPathLength(graph, "A", "C")).toBe(Infinity);
+  });
+
+  it("checks graph connectivity", () => {
+    const connectedGraph = buildGraph({
+      A: { id: "A", q: 0, r: 0, neighbors: ["B"] },
+      B: { id: "B", q: 1, r: 0, neighbors: ["A", "C"] },
+      C: { id: "C", q: 2, r: 0, neighbors: ["B"] },
+    });
+
+    const disconnectedGraph = buildGraph({
+      A: { id: "A", q: 0, r: 0, neighbors: ["B"] },
+      B: { id: "B", q: 1, r: 0, neighbors: ["A"] },
+      C: { id: "C", q: 5, r: 5, neighbors: [] },
+    });
+
+    expect(isConnected(connectedGraph)).toBe(true);
+    expect(isConnected(disconnectedGraph)).toBe(false);
+  });
+});