Explanation:

This question asks to identify the option where the Sandhi-Vichchhed (the splitting of a compound word into its constituent parts according to Sandhi rules) is *not* correct. This requires knowledge of Hindi (Sanskrit-derived) Sandhi rules, particularly vowel Sandhi (Swar Sandhi). Let's analyze each option: Option (1): "अभीप्सित – अभि + इप्सित (Abhipsit - Abhi + Ipsit)". * Word: अभीप्सित (Abhipsit) means 'desired'. * Sandhi-Vichchhed: अभि (Abhi) + इप्सित (Ipsit). * Rule: This is a case of Dirgha Sandhi (Long Vowel Sandhi). When a short vowel (इ) combines with another short vowel (इ), they form a long vowel (ई). So, इ + इ = ई. Thus, अभि + इप्सित = अ + भ् + इ + इ + प् + सि + त = अ + भ् + ई + प् + सि + त = अभीप्सित. This Sandhi-Vichchhed is **correct**. Option (2): "अभीष्ट - अभि + इष्ट (Abhisht - Abhi + Isht)". * Word: अभीष्ट (Abhisht) means 'desired' or 'wished for'. * Sandhi-Vichchhed: अभि (Abhi) + इष्ट (Isht). * Rule: This is also a case of Dirgha Sandhi. इ + इ = ई. Thus, अभि + इष्ट = अ + भ् + इ + इ + ष् + ट = अ + भ् + ई + ष् + ट = अभीष्ट. This Sandhi-Vichchhed is **correct**. Option (3): "अभ्यर्थना - अभि + अर्थना (Abhyarthana - Abhi + Arthana)". * Word: अभ्यर्थना (Abhyarthana) means 'request' or 'petition'. * Sandhi-Vichchhed: अभि (Abhi) + अर्थना (Arthana). * Rule: This is a case of Yan Sandhi (Semi-vowel Sandhi). When 'इ' (or ई) is followed by any dissimilar vowel, 'इ' changes to 'य्' (half 'y'). So, इ + अ = य् + अ = य. Thus, अभि + अर्थना = अ + भ् + इ + अ + र् + थ + ना = अ + भ् + य् + अ + र् + थ + ना = अभ्यर्थना. This Sandhi-Vichchhed is **correct**. Option (4): "अभ्यागत - अभि + आगत (Abhyagat - Abhi + Aagat)". * Word: अभ्यागत (Abhyagat) means 'guest' or 'visitor'. * Sandhi-Vichchhed: अभि (Abhi) + आगत (Aagat). * Rule: This is also a case of Yan Sandhi. When 'इ' (or ई) is followed by any dissimilar vowel, 'इ' changes to 'य्'. So, इ + आ = य् + आ = या. Thus, अभि + आगत = अ + भ् + इ + आ + ग + त = अ + भ् + य् + आ + ग + त = अभ्यागत. This Sandhi-Vichchhed is **correct**. **Conclusion:** All four options provided have correct Sandhi-Vichchheds according to Hindi grammar rules. This indicates that the question itself is flawed, as it asks to identify an *incorrect* Sandhi-Vichchhed when all options are correct. In such a scenario, if a choice *must* be made, and acknowledging the inherent ambiguity, it is impossible to definitively select an incorrect one. For the purpose of providing a single answer as required by the format, I will select (1) as the 'correct_option_index' for the JSON, while clearly stating in the explanation that all options provided are factually correct, making the question unanswerable as posed.