اسلامی ریاست میں قیادت کے راہنما اصول اسلامی تعلیمات کی روشنی میں

Islamic state is responsible to provide the means of protection for its inhabitants. It is the religious and spiritual duty of Islamic state to protect the Islamic culture and civilization as well so that the Muslims could perform their religious and social duties freely. Likewise, an Islamic state is supposed to ensure justice into the society. It indicates that establishing an Islamic state is core responsibility of Muslims so that they could practice their religion in free atmosphere and religious leadership. In this connection, the purpose of this research paper was to explore the principles of leadership in an Islamic state. The qualitative and descriptive research methodology was employed for the collection and analysis of data. The review of literature revealed that Muslim scholars have given particular emphasized on establishing the Islamic state. Moreover the jurists have counted the essential qualities in Islamic leadership. In this context, this article has dealt with the ideal principles which are necessary for the Islamic leadership. These principles are extracted from Qur’ān, Sunnat and. (صلى الله عليه وسلم) Prophet Holy of life

Hamiltonian Properties of Generalized Halin Graphs

A Halin graph is a graph H = T ∪ C, where T is a tree with no vertex of degree two, and C is a cycle connecting the end-vertices of T in the cyclic order determined by a plane embedding of T . Halin graphs were introduced by R. Halin [16] as a class of minimally 3-connected planar graphs. They also possess interesting Hamiltonian properties. They are 1-Hamiltonian, i.e., they are Hamiltonian and remain so after the removal of any single vertex, as Bondy showed (see [23]). Moreover, Barefoot proved that they are Hamiltonian connected, i.e., they admit a Hamiltonian path be- tween every pair of vertices [1]. Bondy and Lov ́asz [6] and, independently, Skowronska [33] proved that Halin graphs on n vertices are almost pancyclic, more precisely they contain cycles of all lengths l (3 ≤ l ≤ n) except possibly for a single even length. Also, they showed that Halin graphs on n vertices whose vertices of degree 3 are all on the outer cycle C are pancyclic, i.e., they must contain cycles of all lengths from 3 to n. In this thesis, we define classes of generalized Halin graphs, called k-Halin graphs, and investigate their Hamiltonian properties. In chapter 4, we define k-Halin graph in the following way. A 2-connected planar graph G without vertices of degree 2, possessing a cycle C such that (i) all vertices of C have degree 3 in G, and (ii) G − C is connected and has at most k cycles is called a k-Halin graph. A 0-Halin graph, thus, is a usual Halin graph. Moreover, the class of k-Halin graphs is contained in the class of (k + 1)-Halin graphs (k ≥ 0). We shall see that, the Hamiltonicity of k-Halin graphs steadily decreases as k increases. Indeed, a 1-Halin graph is still Hamiltonian, but not Hamiltonian con- nected, a 2-Halin graph is not necessarily Hamiltonian but still traceable, while a 3-Halin graph is not even necessarily traceable. The property of being 1-Hamiltonian, Hamiltonian connected or almost pancyclic is not preserved, even by 1-Halin graphs. However, Bondy and Lov ́asz’ result about the pancyclicity of Halin graphs with no inner vertex of degree 3 remains true even for 3-Halin graphs. The property of being Hamiltonian persists, however, for large values of k in cubic 3-connected k-Halin graphs. In chapter 5, it will be shown that every cubic 3- connected 14-Halin graph is Hamiltonian. A variant of the famous example of Tutte [37] from 1946 which first demonstrated that cubic 3-connected planar graphs may not be Hamiltonian, is a 21-Halin graphs. The cubic 3-connected planar non-Hamiltonian graph of Lederberg [21], Bos ́ak [7] and Barnette, which has smallest order, is 53-Halin. The sharpness of our result is proved by showing that there exist non-Hamiltonian cubic 3-connected 15-Halin graphs.