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For a connected graph G = (V, E), an edge monophonic set of G is a set M ⊆ V(G) such that every edge of G is contained in a monophonic path joining some pair of vertices in M. The edge monophonic number m 1 (G) of G is the minimum order of its edge monophonic sets and any edge monophonic set of order m 1 (G) is a minimum edge monophonic set of G. Connected graphs of order p with edge monophonic number p are characterized. Necessary condition for edge monophonic number to be p − 1 is given. It is shown that for every two integers a and b such that 2 ≤ a ≤ b, there exists a connected graph G with m(G) = a and m 1 (G) = b, where m(G) is the monophonic number of G.

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--Optical Character Recognition plays an important role in Digital Image Processing and Pattern Recognition. Even though ambient study had been performed on foreign languages like Chinese and Japanese, effort on Indian script is still immature. OCR in Malayalam language is more complex as it is enriched with largest number of characters among all Indian languages. The challenge of recognition of characters is even high in handwritten domain, due to the varying writing style of each individual. In this paper we propose a system for recognition of offline handwritten Malayalam vowels. The proposed method uses Chain code and Image Centroid for the purpose of extracting features and a two layer feed forward network with scaled conjugate gradient for classification.

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Let G D .V; E/ be a connected graph with at least three vertices. For vertices u and v in G; the distance d.u; v/ is the length of a shortest u v path in G: A u v path of length d.u; v/ is called a u v geodesic. For subsets A and B of V; the distance d.A; B/; is defined as d.A; B/ D mi n fd.x; y/ W x 2 A; y 2 Bg. A u v path of length d.A; B/ is called an A B geodesic joining the sets A; B Â V; where u 2 A and v 2 B: A vertex x is said to lie on an A B geodesic if x is a vertex of an A B geodesic. A set S Â E is called an edge-to-vertex geodetic set if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S: The edge-to-vertex geodetic number g ev .G/ of G is the minimum cardinality of its edge-to-vertex geodetic sets and any edge-to-vertex geodetic set of cardinality g ev .G/ is an edge-to-vertex geodetic basis of G: Any edge-to-vertex geodetic basis is also called a g ev-set of G: It is shown that if G is a connected graph of size q and diameter d; then g ev .G/ Ä q d C 2: It is proved that, for a tree T with q 2; g ev .T / D q d C 2 if and only if T is a caterpillar. For positive integers r; d and l 2 with r Ä d Ä 2r; there exists a connected graph G with rad G D r; d iam G D d and g ev .G/ D l: Also graphs G for which g ev .G/ D q; q 1 or q 2 are characterized.

For a non-trivial connected graph G, a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected edge geodetic set of G is the connected edge geodetic number of G and is denoted by g1c(G). A connected edge geodetic set of cardinality g1c(G) is called a g1c-set of G or connected edge geodetic basis of G. A connected edge geodetic set S in a connected graph G is called a minimal connected edge geodetic set if no proper subset of S is a connected edge geodetic set of G. The upper connected edge geodetic number g + 1c (G) is the maximum cardinality of a minimal connected edge geodetic set of G. Graphs G of order p for which g1c(G) = g + 1c = p are characterized. For positive integers r,d and n ≥ d + 1 with r ≤ d ≤ 2r, there exists a connected graph of radius r, diameter d and upper connected edge geodetic number n. It is shown for any positive integers 2 ≤ a < b ≤ c, there exists a connected graph G such that g1(G) = a, g1c(G) = b and g + 1c (G) = c.

Abstract-In this paper, we propose a handwritten character recognition system for Malayalam language. The feature extraction phase consists of gradient and curvature calculation and dimensionality reduction using Principal Component Analysis. Directional information from the arc tangent of gradient is used as gradient feature. Strength of gradient in curvature direction is used as the curvature feature. The proposed system uses a combination of gradient and curvature feature in reduced dimension as the feature vector. For classification, discriminative power of Support Vector Machine (SVM) is evaluated. The results reveal that SVM with Radial Basis Function (RBF) kernel yield the best performance with 96.28% and 97.96% of accuracy in two different datasets. This is the highest accuracy ever reported on these datasets.