feat: added TwoSat solutions with Tests, also updated Directory.md (#6756)

* feat: added TwoSat solutions and Tests and also updated Directory.md

* added reference urls to the article related to the 2-SAT

* Fix generic type warnings in TwoSat.java

* maven build fix: added rawtypes in SuppressWarnings

* fixed  checkstyle error

* fix: Sorted import order

---------

Co-authored-by: a <alexanderklmn@gmail.com>

src/test/java/com/thealgorithms/datastructures/graphs/TwoSatTest.java

@@ -0,0 +1,125 @@
+package com.thealgorithms.datastructures.graphs;
+
+import static org.junit.jupiter.api.Assertions.assertArrayEquals;
+import static org.junit.jupiter.api.Assertions.assertFalse;
+import static org.junit.jupiter.api.Assertions.assertTrue;
+
+import org.junit.jupiter.api.Test;
+
+/**
+ * Testcases for 2-SAT.
+ * Please note thea whlie checking for boolean assignments always keep n + 1 elements and the first element should be always false.
+ */
+public class TwoSatTest {
+    private TwoSat twoSat;
+
+    /**
+     * Case 1: Basic satisfiable case.
+     * Simple 3 clauses with consistent assignments.
+     */
+    @Test
+    public void testSatisfiableBasicCase() {
+        twoSat = new TwoSat(5);
+
+        twoSat.addClause(1, false, 2, false); // (x1 ∨ x2)
+        twoSat.addClause(3, true, 2, false); // (¬x3 ∨ x2)
+        twoSat.addClause(4, false, 5, true); // (x4 ∨ ¬x5)
+
+        twoSat.solve();
+
+        assertTrue(twoSat.isSolutionExists(), "Expected solution to exist");
+        boolean[] expected = {false, true, true, true, true, true};
+        assertArrayEquals(expected, twoSat.getSolutions());
+    }
+
+    /**
+     * Case 2: Unsatisfiable due to direct contradiction.
+     * (x1 ∨ x1) ∧ (¬x1 ∨ ¬x1) makes x1 and ¬x1 both required.
+     */
+    @Test
+    public void testUnsatisfiableContradiction() {
+        twoSat = new TwoSat(1);
+
+        twoSat.addClause(1, false, 1, false); // (x1 ∨ x1)
+        twoSat.addClause(1, true, 1, true); // (¬x1 ∨ ¬x1)
+
+        twoSat.solve();
+
+        assertFalse(twoSat.isSolutionExists(), "Expected no solution (contradiction)");
+    }
+
+    /**
+     * Case 3: Single variable, trivially satisfiable.
+     * Only (x1 ∨ x1) exists.
+     */
+    @Test
+    public void testSingleVariableTrivialSatisfiable() {
+        twoSat = new TwoSat(1);
+
+        twoSat.addClause(1, false, 1, false); // (x1 ∨ x1)
+
+        twoSat.solve();
+
+        assertTrue(twoSat.isSolutionExists(), "Expected solution to exist");
+        boolean[] expected = {false, true};
+        assertArrayEquals(expected, twoSat.getSolutions());
+    }
+
+    /**
+     * Case 4: Larger satisfiable system with dependencies.
+     * (x1 ∨ x2), (¬x2 ∨ x3), (¬x3 ∨ x4), (¬x4 ∨ x5)
+     */
+    @Test
+    public void testChainedDependenciesSatisfiable() {
+        twoSat = new TwoSat(5);
+
+        twoSat.addClause(1, false, 2, false);
+        twoSat.addClause(2, true, 3, false);
+        twoSat.addClause(3, true, 4, false);
+        twoSat.addClause(4, true, 5, false);
+
+        twoSat.solve();
+
+        assertTrue(twoSat.isSolutionExists(), "Expected solution to exist");
+        boolean[] solution = twoSat.getSolutions();
+        for (int i = 1; i <= 5; i++) {
+            assertTrue(solution[i], "Expected x" + i + " to be true");
+        }
+    }
+
+    /**
+     * Case 5: Contradiction due to dependency cycle.
+     * (x1 ∨ x2), (¬x1 ∨ ¬x2), (x1 ∨ ¬x2), (¬x1 ∨ x2)
+     * These clauses form a circular dependency making it impossible.
+     */
+    @Test
+    public void testUnsatisfiableCycle() {
+        twoSat = new TwoSat(2);
+
+        twoSat.addClause(1, false, 2, false);
+        twoSat.addClause(1, true, 2, true);
+        twoSat.addClause(1, false, 2, true);
+        twoSat.addClause(1, true, 2, false);
+
+        twoSat.solve();
+
+        assertFalse(twoSat.isSolutionExists(), "Expected no solution due to contradictory cycle");
+    }
+
+    /**
+     * Testcase from CSES
+     */
+    @Test
+    public void test6() {
+        twoSat = new TwoSat(2);
+
+        twoSat.addClause(1, true, 2, false);
+        twoSat.addClause(2, true, 1, false);
+        twoSat.addClause(1, true, 1, true);
+        twoSat.addClause(2, false, 2, false);
+
+        twoSat.solve();
+
+        assertFalse(twoSat.isSolutionExists(), "Expected no solution.");
+    }
+}