THE DETOUR MONOPHONIC GRPHOIDAL COVERING NUMBER OF A GRAPH

P. TITUS,S. SANTHA KUMARI 장전수학회 2016 Proceedings of the Jangjeon mathematical society Vol.19 No.1

A chord of a path P is an edge joining two non-adjacent ver- tices of P. A path P is called a monophonic path if it is a chordless path. A path P is called a detour monophonic path in G if it is a longest mono- phonic path in G. A detour monophonic graphoidal cover of a graph G is a collection dm of detour monophonic paths in G such that every vertex of G is an internal vertex of at most one detour monophonic path in dm and every edge of G is in exactly one detour monophonic path in dm. The minimum cardinality of a detour monophonic graphoidal cover of G is called the detour monophonic graphoidal covering num- ber of G and is denoted by dm(G). We determine bounds for it and characterize graphs which realize the bounds. Also, we find the detour monophonic graphoidal covering number of unicyclic graphs.

The forcing connected vertex monophonic number of a graph

P. Titus,K. IYAPPAN 장전수학회 2016 Advanced Studies in Contemporary Mathematics Vol.26 No.1

For any vertex x in a connected graph G of order p ≥ 2, a set Sx ⊆ V(G) is an x-monophonic set of G if each vertex v ∈ V(G) lies on an x - y monophonic path for some element y in Sx. The minimum cardinality of an x-monophonic set of g is the x-monophonic set of G is an x-monophonic set Sx such that the subgraph G[Sx]induced by Sx is connected. The minimum cardinality of a connected x-monophonic set of G is defined as the connected x-monophonic number of G and is denoted by cmx(G). A subset T of a minimum connected x-monophonic set Sx of G is called an x-forcing subset for Sx if Sx is the unique minimum connected x-monophonic set containing T. An x-forcing subset for Sx of minimum cardinality is a minimum x-forcing subset of Sx. The forcing connected x-monophonic number of Sx, denoted by fcmx(Sx), is the cardinality of a minimum x-forcing subset for Sx. The forcing connected x-monophonic number of G is fcmx (G) = min {fcmx (Sx)}, where the minimum is taken over all minimum connected x-monophonic sets Sx in G. We determine bounds for it and find the forcing connected vertex monophonic number for some special classes of graphs. For any two positive integers a and b with 0≤ a < b-1, there exists a connected graph G with fcmx(G) = a and cmx (G) =b for some vertex x in G.

Prophetic Ways of Transmitting the Word of God

P. Joseph Titus(P. 요셉 타이투스) 신학과사상학회 2015 가톨릭 신학과 사상 Vol.- No.76

Connected edge detour monophonic number of a graph

P. Titus,P. Balakrishnan,A. P. Santhakumaran,K. Ganesamoorthy 장전수학회 2013 Proceedings of the Jangjeon mathematical society Vol.16 No.4

Connected edge detour monophonic number of a graph

하느님 말씀을 전달하는 예언자적 방식

P. 요셉 타이투스 신부(Fr.P. Joseph Titus) 신학과사상학회 2015 가톨릭 신학과 사상 Vol.- No.76

This article titled “Prophetic Ways of Transmitting the Word of God” aims at expounding the role of the Israelite prophets in communicating the Word of God to the people of their times. The purpose of this venture is to draw insights from the prophets to communicate God’s Word to the people of our times. This article consists of three parts. The first part tries to show that by their vocation, Israelite prophets were regarded primarily as official recipients and transmitters of the Word of the Lord (Jer 1:2), for they hear and speak what the Lord has spoken to them (Isa 1:2; 6:8-10). This role of the prophets is very well brought out in Deut 18:18, wherein the Lord himself says: “I will put my words in the mouth of the prophet, who shall speak to them everything that I command.” This is why the people of Israel expected from the prophets the word (dabar), as they expected instruction (torah) from the priest and counsel (esah) from the wise (Jer 18:18). Therefore if there is no prophet in Israel, there would be “famine of hearing the Word of God” (Am 8:1). The second part discusses the manner in which the prophets carried out their ministry of the Word of God. God entrusted the prophets with his Word not to keep it to themselves but to transmit it to the people. The article argues that the prophets have attempted to perform their ministry of God’s Word through at least four modes, namely through their words, deeds, life and writings. First of all, the prophet is a man of the Word. Whatever he speaks is initiated by God. As he is God’s messenger, he begins always his speech in the divine first person: “Thus says the Lord.” Secondly, prophetic message is more than words. The prophets drama tized their spoken word with symbolic action. A number of reports of symbolic action could be found in the prophetic literature. Thirdly, at times the very life of the prophets of the OT conveyed God’s Word to the people: the unhappy marriage of Hosea, the celibacy of Jeremiah, his exclusion from the Temple, the catalepsy of Ezekiel were still more eloquent than their words and symbolical actions. Finally the prophets transmitted God’s Word not only orally and by their very life, but also through writing scrolls. In its third part, the article explains the sufferings encountered by the prophets while delivering God’s Word. During their ministry, the prophets risked their reputations and their lives by challenging the accepted standards of society. Because of their suffering, at times, some of them felt the temptation to abandon their ministry. The article concludes by inviting us to appreciate the prophetic imagination and creativity which they employed to adapt alternative means of communication in the ministry. In order to make a memorable impression of God’s Word in the minds of our audience, we are invited to discern the most fitting model of transmission for the appropriate situation for our times.

The upper connected vertex monophonic number of a graph

P. Titus 장전수학회 2014 Proceedings of the Jangjeon mathematical society Vol.17 No.2

The upper connected vertex monophonic number of a graph

ON THE MONOPHONIC NUMBER OF A GRAPH

A. P. Santhakumaran,P. Titus,K. Ganesamoorthy 한국전산응용수학회 2014 Journal of applied mathematics & informatics Vol.32 No.1

For a connected graph G = (V,E) of order at least two, a set S of vertices of G is a monophonic set of G if each vertex v of G lies on an χ−y monophonic path for some elements χ and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G, denoted by m(G). Certain general properties satisfied by the monophonic sets are studied. Graphs G of order p with m(G) = 2 or p or p −1 are characterized. For every pair a, b of positive integers with 2≤a≤b, there is a connected graph G with m(G) = a and g(G) = b, where g(G) is the geodetic number of G. Also we study how the monophonic number of a graph is affected when pendant edges are added to the graph.

Upper edge-to-vertex detour monophonic number of a graph

A. P. Santhakumaran,P. Titus,K. Ganesamoorthy 장전수학회 2016 Advanced Studies in Contemporary Mathematics Vol.26 No.2

For a connected graph G = (V,E) of order at least three, the monophonic distance dm(u, v) is the length of a longest u−v monophonic path in G. A u − v path of length dm(u, v) is called a u − v detour monophonic. For subsets A and B of V , the monophonic distance dm(A,B) is defined as dm(A,B) = min{dm(x, y) : x ∈ A, y ∈ B}. A u−v path of length dm(A, B) is called an A−B detour monophonic path joining the sets A,B ⊆ V, where u ∈ A and v ∈ B. A set S ⊆ E is called an edge-to-vertex detour monophonic set of G if every vertex of G is incident with an edge of S or lies on a detour monophonic path joining a pair of edges of S. The edge-to-vertex detour monophonic number dmev(G) of G is the minimum cardinality of its edge-to-vertex detour monophonic sets and any edge-to-vertex detour monophonic set of car- dinality dmev(G) is an edge-to-vertex detour monophonic basis of G. An edge-to-vertex detour monophonic set S in a connected graph G is called a minimal edge-to-vertex detour monophonic set of G if no proper subset of S is an edge-to-vertex detour monophonic set of G. The upper edge-to-vertex detour monophonic number dm+ ev(G) of G is the maxi- mum cardinality of a minimal edge-to-vertex detour monophonic set of G. We determine bounds for it and certain general properties of these concepts are studied. It is shown that for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G with dmev(G) = a and dm+ ev(G) = b.

ON THE MONOPHONIC NUMBER OF A GRAPH

Santhakumaran, A.P.,Titus, P.,Ganesamoorthy, K. The Korean Society for Computational and Applied M 2014 Journal of applied mathematics & informatics Vol.32 No.1

For a connected graph G = (V,E) of order at least two, a set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x - y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G, denoted by m(G). Certain general properties satisfied by the monophonic sets are studied. Graphs G of order p with m(G) = 2 or p or p - 1 are characterized. For every pair a, b of positive integers with $2{\leq}a{\leq}b$, there is a connected graph G with m(G) = a and g(G) = b, where g(G) is the geodetic number of G. Also we study how the monophonic number of a graph is affected when pendant edges are added to the graph.

Results of availability imposed configuration details developed for K-DEMO

Brown, T.,Titus, P.,Brooks, A.,Zhang, H.,Neilson, H.,Im, K.,Kim, K. North-Holland ; Elsevier Science Ltd 2016 Fusion engineering and design Vol.109 No.2

The Korean fusion demonstration reactor (K-DEMO) has completed a two year study looking at key Tokamak components and configuration options in preparation of a conceptual design phase. A key part of a device configuration centers on defining an arrangement that enhances the ability to reach high availability values by defining design solutions that foster simplified maintenance operations. To maximize the size and minimize the number of in-vessel components enlarged TF coils were defined that incorporate a pair of windings within each coil to mitigate pressure drop issues and to reduce the cost of the coils. A semi-permanent shield structure was defined to develop labyrinth interfaces between double-null plasma contoured shield modules, provide an entity to align blanket components and provide support against disruption loads-with a load path that equilibrates blanket, TF and PF loads through a base structure. Blanket piping services and auxiliary systems that interface with in-vessel components have played a major role in defining the overall device arrangement-concept details will be presented along with general arrangement features and preliminary results obtained from disruption analysis.