diff --git a/build.xml b/build.xml
index 1e5f6542..1776a551 100755
--- a/build.xml
+++ b/build.xml
@@ -74,6 +74,12 @@
             <include name="ShortestPathAndMSTTest.java"/>
             <include name="SubdividedGraphTest.java"/>
             <include name="CoronaAndUtilsTest.java"/>
+            <include name="BiconnectedComponentsTest.java"/>
+            <include name="G6FormatTest.java"/>
+            <include name="GraphComplementTest.java"/>
+            <include name="GraphJoinTest.java"/>
+            <include name="RenderTableTest.java"/>
+            <include name="UtilityAndCheckerTest.java"/>
         </javac>
         <junitlauncher haltOnFailure="false" printSummary="true">
             <classpath>
@@ -107,6 +113,12 @@
                     <include name="ShortestPathAndMSTTest.class"/>
                     <include name="SubdividedGraphTest.class"/>
                     <include name="CoronaAndUtilsTest.class"/>
+                    <include name="BiconnectedComponentsTest.class"/>
+                    <include name="G6FormatTest.class"/>
+                    <include name="GraphComplementTest.class"/>
+                    <include name="GraphJoinTest.class"/>
+                    <include name="RenderTableTest.class"/>
+                    <include name="UtilityAndCheckerTest.class"/>
                 </fileset>
                 <listener type="legacy-xml" outputdir="${basedir}/build/test-classes" sendSysOut="true" sendSysErr="true"/>
                 <listener type="legacy-plain" sendSysOut="true"/>
diff --git a/test/UtilityAndCheckerTest.java b/test/UtilityAndCheckerTest.java
new file mode 100644
index 00000000..37c78cbd
--- /dev/null
+++ b/test/UtilityAndCheckerTest.java
@@ -0,0 +1,358 @@
+import graphtea.extensions.algorithms.GOpUtils;
+import graphtea.extensions.generators.CircleGenerator;
+import graphtea.extensions.generators.CompleteGraphGenerator;
+import graphtea.extensions.generators.PathGenerator;
+import graphtea.graph.graph.Edge;
+import graphtea.graph.graph.GPoint;
+import graphtea.graph.graph.GraphModel;
+import graphtea.graph.graph.Vertex;
+import graphtea.library.algorithms.util.AcyclicChecker;
+import graphtea.library.algorithms.util.BipartiteChecker;
+import graphtea.library.algorithms.util.ConnectivityChecker;
+import graphtea.library.util.Pair;
+import org.junit.jupiter.api.Test;
+
+import java.util.HashMap;
+
+import static org.junit.jupiter.api.Assertions.*;
+
+/**
+ * Tests for utility classes and graph-property checkers that previously had no
+ * dedicated test coverage:
+ *   - AcyclicChecker
+ *   - BipartiteChecker
+ *   - ConnectivityChecker
+ *   - GOpUtils
+ *   - Pair
+ */
+public class UtilityAndCheckerTest {
+
+    // =========================================================================
+    // AcyclicChecker
+    // =========================================================================
+
+    /** A single isolated vertex has no edges, so it is a tree (acyclic). */
+    @Test
+    public void testAcyclicCheckerSingleVertex() {
+        GraphModel g = new GraphModel(false);
+        g.insertVertex(new Vertex());
+        assertTrue(AcyclicChecker.isGraphAcyclic(g));
+    }
+
+    /** A path graph is acyclic (it is a tree). */
+    @Test
+    public void testAcyclicCheckerPathIsAcyclic() {
+        GraphModel g = PathGenerator.generatePath(5);
+        assertTrue(AcyclicChecker.isGraphAcyclic(g));
+    }
+
+    /** A cycle graph C_n (n ≥ 3) contains a cycle, so it is not acyclic. */
+    @Test
+    public void testAcyclicCheckerCycleIsNotAcyclic() {
+        GraphModel g = CircleGenerator.generateCircle(4);
+        assertFalse(AcyclicChecker.isGraphAcyclic(g));
+    }
+
+    /** A complete graph K_4 has many cycles and is not acyclic. */
+    @Test
+    public void testAcyclicCheckerCompleteK4IsNotAcyclic() {
+        GraphModel g = CompleteGraphGenerator.generateCompleteGraph(4);
+        assertFalse(AcyclicChecker.isGraphAcyclic(g));
+    }
+
+    /** Two isolated vertices (no edges) form a forest — acyclic. */
+    @Test
+    public void testAcyclicCheckerTwoIsolatedVerticesIsAcyclic() {
+        GraphModel g = new GraphModel(false);
+        g.insertVertex(new Vertex());
+        g.insertVertex(new Vertex());
+        assertTrue(AcyclicChecker.isGraphAcyclic(g));
+    }
+
+    /** A single edge (K_2) is a tree — acyclic. */
+    @Test
+    public void testAcyclicCheckerSingleEdgeIsAcyclic() {
+        GraphModel g = PathGenerator.generatePath(2);
+        assertTrue(AcyclicChecker.isGraphAcyclic(g));
+    }
+
+    /** A triangle (C_3) is not acyclic. */
+    @Test
+    public void testAcyclicCheckerTriangleIsNotAcyclic() {
+        GraphModel g = CircleGenerator.generateCircle(3);
+        assertFalse(AcyclicChecker.isGraphAcyclic(g));
+    }
+
+    // =========================================================================
+    // BipartiteChecker
+    // =========================================================================
+
+    /** A path graph is always bipartite (its two colour-classes alternate). */
+    @Test
+    public void testBipartiteCheckerPathIsBipartite() {
+        GraphModel g = PathGenerator.generatePath(5);
+        assertTrue(BipartiteChecker.isBipartite(g));
+    }
+
+    /** An even cycle C_4 is bipartite. */
+    @Test
+    public void testBipartiteCheckerEvenCycleIsBipartite() {
+        GraphModel g = CircleGenerator.generateCircle(4);
+        assertTrue(BipartiteChecker.isBipartite(g));
+    }
+
+    /** A larger even cycle C_6 is bipartite. */
+    @Test
+    public void testBipartiteCheckerC6IsBipartite() {
+        GraphModel g = CircleGenerator.generateCircle(6);
+        assertTrue(BipartiteChecker.isBipartite(g));
+    }
+
+    /** An odd cycle C_3 (triangle) is NOT bipartite. */
+    @Test
+    public void testBipartiteCheckerOddCycleC3IsNotBipartite() {
+        GraphModel g = CircleGenerator.generateCircle(3);
+        assertFalse(BipartiteChecker.isBipartite(g));
+    }
+
+    /** An odd cycle C_5 is NOT bipartite. */
+    @Test
+    public void testBipartiteCheckerOddCycleC5IsNotBipartite() {
+        GraphModel g = CircleGenerator.generateCircle(5);
+        assertFalse(BipartiteChecker.isBipartite(g));
+    }
+
+    /** K_4 (complete on 4 vertices) contains odd cycles, so it is NOT bipartite. */
+    @Test
+    public void testBipartiteCheckerCompleteK4IsNotBipartite() {
+        GraphModel g = CompleteGraphGenerator.generateCompleteGraph(4);
+        assertFalse(BipartiteChecker.isBipartite(g));
+    }
+
+    /** K_2 (single edge) is bipartite — the two endpoints form the two parts. */
+    @Test
+    public void testBipartiteCheckerK2IsBipartite() {
+        GraphModel g = CompleteGraphGenerator.generateCompleteGraph(2);
+        assertTrue(BipartiteChecker.isBipartite(g));
+    }
+
+    /** A single vertex with no edges is trivially bipartite. */
+    @Test
+    public void testBipartiteCheckerSingleVertexIsBipartite() {
+        GraphModel g = new GraphModel(false);
+        g.insertVertex(new Vertex());
+        assertTrue(BipartiteChecker.isBipartite(g));
+    }
+
+    // =========================================================================
+    // ConnectivityChecker
+    // =========================================================================
+
+    /**
+     * An empty graph (no vertices) is vacuously connected: the implementation
+     * iterates over vertices and immediately returns true when there are none,
+     * consistent with the convention that the empty graph satisfies all
+     * universal (for-all) properties.
+     */
+    @Test
+    public void testConnectivityCheckerEmptyGraphIsConnected() throws Exception {
+        GraphModel g = new GraphModel(false);
+        assertTrue(ConnectivityChecker.isGraphConnected(g));
+    }
+
+    /** A single vertex is connected. */
+    @Test
+    public void testConnectivityCheckerSingleVertexIsConnected() throws Exception {
+        GraphModel g = new GraphModel(false);
+        g.insertVertex(new Vertex());
+        assertTrue(ConnectivityChecker.isGraphConnected(g));
+    }
+
+    /** A path graph P_5 is connected. */
+    @Test
+    public void testConnectivityCheckerPathIsConnected() throws Exception {
+        GraphModel g = PathGenerator.generatePath(5);
+        assertTrue(ConnectivityChecker.isGraphConnected(g));
+    }
+
+    /** A cycle C_4 is connected. */
+    @Test
+    public void testConnectivityCheckerCycleIsConnected() throws Exception {
+        GraphModel g = CircleGenerator.generateCircle(4);
+        assertTrue(ConnectivityChecker.isGraphConnected(g));
+    }
+
+    /** K_4 is connected. */
+    @Test
+    public void testConnectivityCheckerCompleteK4IsConnected() throws Exception {
+        GraphModel g = CompleteGraphGenerator.generateCompleteGraph(4);
+        assertTrue(ConnectivityChecker.isGraphConnected(g));
+    }
+
+    /** Two isolated vertices (no edges) form a disconnected graph. */
+    @Test
+    public void testConnectivityCheckerTwoIsolatedVerticesIsDisconnected() throws Exception {
+        GraphModel g = new GraphModel(false);
+        g.insertVertex(new Vertex());
+        g.insertVertex(new Vertex());
+        assertFalse(ConnectivityChecker.isGraphConnected(g));
+    }
+
+    /** A graph with two components (two separate edges) is not connected. */
+    @Test
+    public void testConnectivityCheckerTwoSeparateEdgesIsDisconnected() throws Exception {
+        GraphModel g = new GraphModel(false);
+        Vertex v0 = new Vertex();
+        Vertex v1 = new Vertex();
+        Vertex v2 = new Vertex();
+        Vertex v3 = new Vertex();
+        g.insertVertex(v0);
+        g.insertVertex(v1);
+        g.insertVertex(v2);
+        g.insertVertex(v3);
+        g.insertEdge(new Edge(v0, v1));
+        g.insertEdge(new Edge(v2, v3));
+        assertFalse(ConnectivityChecker.isGraphConnected(g));
+    }
+
+    // =========================================================================
+    // GOpUtils
+    // =========================================================================
+
+    /** center() of a single vertex at (3, 4) should be (3, 4). */
+    @Test
+    public void testGOpUtilsCenterSingleVertex() {
+        GraphModel g = new GraphModel(false);
+        Vertex v = new Vertex();
+        v.setLocation(new GPoint(3.0, 4.0));
+        g.insertVertex(v);
+
+        GPoint c = GOpUtils.center(g);
+        assertEquals(3.0, c.x, 1e-9);
+        assertEquals(4.0, c.y, 1e-9);
+    }
+
+    /** center() of two vertices at (0,0) and (4,6) should be (2, 3). */
+    @Test
+    public void testGOpUtilsCenterTwoVertices() {
+        GraphModel g = new GraphModel(false);
+        Vertex v1 = new Vertex(); v1.setLocation(new GPoint(0.0, 0.0));
+        Vertex v2 = new Vertex(); v2.setLocation(new GPoint(4.0, 6.0));
+        g.insertVertex(v1); g.insertVertex(v2);
+
+        GPoint c = GOpUtils.center(g);
+        assertEquals(2.0, c.x, 1e-9);
+        assertEquals(3.0, c.y, 1e-9);
+    }
+
+    /** center() of a symmetric arrangement should lie at the origin. */
+    @Test
+    public void testGOpUtilsCenterSymmetricArrangement() {
+        GraphModel g = new GraphModel(false);
+        Vertex v1 = new Vertex(); v1.setLocation(new GPoint(-1.0, 0.0));
+        Vertex v2 = new Vertex(); v2.setLocation(new GPoint(1.0, 0.0));
+        Vertex v3 = new Vertex(); v3.setLocation(new GPoint(0.0, 2.0));
+        Vertex v4 = new Vertex(); v4.setLocation(new GPoint(0.0, -2.0));
+        g.insertVertex(v1); g.insertVertex(v2);
+        g.insertVertex(v3); g.insertVertex(v4);
+
+        GPoint c = GOpUtils.center(g);
+        assertEquals(0.0, c.x, 1e-9);
+        assertEquals(0.0, c.y, 1e-9);
+    }
+
+    /** offsetPositionsToCenter() returns a map with one entry per vertex. */
+    @Test
+    public void testGOpUtilsOffsetPositionsToCenterEntryCount() {
+        GraphModel g = new GraphModel(false);
+        Vertex v1 = new Vertex(); v1.setLocation(new GPoint(1.0, 1.0));
+        Vertex v2 = new Vertex(); v2.setLocation(new GPoint(3.0, 3.0));
+        g.insertVertex(v1); g.insertVertex(v2);
+
+        HashMap<Integer, GPoint> offsets = GOpUtils.offsetPositionsToCenter(g);
+        assertEquals(2, offsets.size());
+    }
+
+    /** offsetPositionsToCenter() offsets sum to zero for a symmetric graph. */
+    @Test
+    public void testGOpUtilsOffsetPositionsToCenterSymmetric() {
+        GraphModel g = new GraphModel(false);
+        Vertex v1 = new Vertex(); v1.setLocation(new GPoint(-1.0, 0.0));
+        Vertex v2 = new Vertex(); v2.setLocation(new GPoint(1.0, 0.0));
+        g.insertVertex(v1); g.insertVertex(v2);
+
+        HashMap<Integer, GPoint> offsets = GOpUtils.offsetPositionsToCenter(g);
+        double sumX = offsets.values().stream().mapToDouble(p -> p.x).sum();
+        double sumY = offsets.values().stream().mapToDouble(p -> p.y).sum();
+        assertEquals(0.0, sumX, 1e-9);
+        assertEquals(0.0, sumY, 1e-9);
+    }
+
+    /** midPoint() of an edge between (0,0) and (4,6) should be (2, 3). */
+    @Test
+    public void testGOpUtilsMidPoint() {
+        GraphModel g = new GraphModel(false);
+        Vertex v1 = new Vertex(); v1.setLocation(new GPoint(0.0, 0.0));
+        Vertex v2 = new Vertex(); v2.setLocation(new GPoint(4.0, 6.0));
+        g.insertVertex(v1); g.insertVertex(v2);
+        Edge e = new Edge(v1, v2);
+        g.insertEdge(e);
+
+        GPoint mid = GOpUtils.midPoint(e);
+        assertEquals(2.0, mid.x, 1e-9);
+        assertEquals(3.0, mid.y, 1e-9);
+    }
+
+    /** midPoint() of an edge between two identical points should equal that point. */
+    @Test
+    public void testGOpUtilsMidPointSameLocation() {
+        GraphModel g = new GraphModel(false);
+        Vertex v1 = new Vertex(); v1.setLocation(new GPoint(5.0, 7.0));
+        Vertex v2 = new Vertex(); v2.setLocation(new GPoint(5.0, 7.0));
+        g.insertVertex(v1); g.insertVertex(v2);
+        Edge e = new Edge(v1, v2);
+        g.insertEdge(e);
+
+        GPoint mid = GOpUtils.midPoint(e);
+        assertEquals(5.0, mid.x, 1e-9);
+        assertEquals(7.0, mid.y, 1e-9);
+    }
+
+    // =========================================================================
+    // Pair
+    // =========================================================================
+
+    /** Pair stores its two values and exposes them via the public fields. */
+    @Test
+    public void testPairHoldsValues() {
+        Pair<String, Integer> p = new Pair<>("hello", 42);
+        assertEquals("hello", p.first);
+        assertEquals(42, p.second);
+    }
+
+    /** Pair allows null values. */
+    @Test
+    public void testPairAllowsNull() {
+        Pair<String, String> p = new Pair<>(null, null);
+        assertNull(p.first);
+        assertNull(p.second);
+    }
+
+    /** Pair fields are mutable (public). */
+    @Test
+    public void testPairFieldsAreMutable() {
+        Pair<Integer, Integer> p = new Pair<>(1, 2);
+        p.first = 10;
+        p.second = 20;
+        assertEquals(10, p.first);
+        assertEquals(20, p.second);
+    }
+
+    /** Pair works with different generic type combinations. */
+    @Test
+    public void testPairGenericTypes() {
+        Pair<Double, Boolean> p = new Pair<>(3.14, true);
+        assertEquals(3.14, p.first, 1e-15);
+        assertTrue(p.second);
+    }
+}