diff --git a/src/game/chase/hexGraph.ts b/src/game/chase/hexGraph.ts
index 6361d78..e4011ca 100644
--- a/src/game/chase/hexGraph.ts
+++ b/src/game/chase/hexGraph.ts
@@ -16,19 +16,21 @@ export function shortestPathLength(
   from: NodeId,
   to: NodeId,
 ): number {
-  if (from === to) {
-    return 0;
-  }
-
   if (!graph.nodes[from] || !graph.nodes[to]) {
     return Infinity;
   }
 
+  if (from === to) {
+    return 0;
+  }
+
   const queue: Array<[NodeId, number]> = [[from, 0]];
+  let head = 0;
   const visited = new Set<NodeId>([from]);
 
-  while (queue.length > 0) {
-    const [current, distance] = queue.shift()!;
+  while (head < queue.length) {
+    const [current, distance] = queue[head];
+    head += 1;
     const node = graph.nodes[current];
 
     for (const neighbor of node.neighbors) {
@@ -54,9 +56,11 @@ export function isConnected(graph: GameGraph): boolean {
   const start = nodeIds[0];
   const visited = new Set<NodeId>([start]);
   const queue: NodeId[] = [start];
+  let head = 0;
 
-  while (queue.length > 0) {
-    const current = queue.shift()!;
+  while (head < queue.length) {
+    const current = queue[head];
+    head += 1;
     const node = graph.nodes[current];
 
     for (const neighbor of node.neighbors) {
diff --git a/tests/chase/hexGraph.test.ts b/tests/chase/hexGraph.test.ts
index d8fd6de..2ee9f4c 100644
--- a/tests/chase/hexGraph.test.ts
+++ b/tests/chase/hexGraph.test.ts
@@ -48,6 +48,17 @@ describe("hex graph utilities", () => {
     expect(shortestPathLength(graph, "A", "C")).toBe(Infinity);
   });
 
+  it("returns Infinity for invalid from/to nodes", () => {
+    const graph = buildGraph({
+      A: { id: "A", q: 0, r: 0, neighbors: ["B"] },
+      B: { id: "B", q: 1, r: 0, neighbors: ["A"] },
+    });
+
+    expect(shortestPathLength(graph, "X", "X")).toBe(Infinity);
+    expect(shortestPathLength(graph, "X", "A")).toBe(Infinity);
+    expect(shortestPathLength(graph, "A", "X")).toBe(Infinity);
+  });
+
   it("checks graph connectivity", () => {
     const connectedGraph = buildGraph({
       A: { id: "A", q: 0, r: 0, neighbors: ["B"] },
@@ -64,4 +75,14 @@ describe("hex graph utilities", () => {
     expect(isConnected(connectedGraph)).toBe(true);
     expect(isConnected(disconnectedGraph)).toBe(false);
   });
+
+  it("treats empty and single-node graph as connected", () => {
+    const emptyGraph = buildGraph({});
+    const singleNodeGraph = buildGraph({
+      A: { id: "A", q: 0, r: 0, neighbors: [] },
+    });
+
+    expect(isConnected(emptyGraph)).toBe(true);
+    expect(isConnected(singleNodeGraph)).toBe(true);
+  });
 });