For any two vertices u and v in a connected graph G, a u – v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) from u to v is defined as the length of a longest u – v monophonic path in G. A u – v monophonic path of length dm(u, v) is called a u – v monophonic. The monophonic eccentricity em(v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius rad m G of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diam m G of G is the maximum monophonic eccentricity among the vertices of G. It is shown that rad m G ≤ diam m G for every connected graph G and that every pair a, b of positive integers with a ≤ b is realizable as the monophonic radius and monophonic diameter of some connected graph. Also, for any three positive integers a, b and c with 3 ≤ a ≤ b ≤ c, there is a connected graph G such that rad G = a, rad m G = b and rad DG = c; and for any three positive integers a, b and c with 5 ≤ a ≤ b ≤ c, there is a connected graph G such that diam G = a, diam m G = b and diam D G = c, where rad G, diam G, rad DG and diam D G denote the radius, diameter, detour radius and detour diameter, respectively. The monophonic center of G is the subgraph induced by the vertices of G having monophonic eccentricity rad m G and it is shown that every graph is the monophonic center of some connected graph and also that the monophonic center Cm(G) of every connected graph G is a subgraph of some block of G.

Purpose To identify factors associated with best-corrected visual acuity (BCVA) presentation and two-year outcome in 479 intermediate, posterior, and panuveitic eyes. Design Cohort study using randomized controlled trial data Methods Multicenter Uveitis Steroid Treatment (MUST) Trial masked BCVA measurements at baseline and 2 years’ follow-up used gold standard methods. Twenty-three clinical centers documented characteristics per protocol, which were evaluated as potential predictive factors for baseline BCVA and two-year change in BCVA. Results Baseline factors significantly associated with reduced BCVA included: age ≥50 vs. <50 years; posterior vs. intermediate uveitis; uveitis duration >10 vs. <6 years; anterior chamber (AC) flare > grade 0; cataract; macular thickening; and exudative retinal detachment. Over two years, eyes better than 20/50 and 20/50 or worse at baseline improved, on average, by 1 letter (p=0.52) and 10 letters (p<0.001) respectively. Both treatment groups and all sites of uveitis improved similarly. Factors associated with improved BCVA included resolution of active AC cells, of macular thickening, and cataract surgery in an initially cataractous eye. Factors associated with worsening BCVA included longer duration of uveitis (6–10 or >10 vs. <6 years), incident AC flare, cataract at both baseline and follow-up, pseudophakia at baseline, persistence or incidence of vitreous haze, and incidence of macular thickening. Conclusions Intermediate, posterior and panuveitis have a similarly favorable prognosis with both systemic and fluocinolone acetonide implant treatment. Eyes with more prolonged/severe inflammatory damage and/or inflammatory findings initially or during follow-up have a worse visual acuity prognosis. The results indicate the value of implementing best practices in managing inflammation.

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 ≤ 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.

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For a connected graph G of order p ≥ 2 and a vertex x of G, a set S ⊆ 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 S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G, denoted by m x (G). An x-monophonic set of cardinality m x (G) is called a m x -set of G. We determine bounds for it and characterize graphs which realize these bounds. A connected graph of order p with vertex monophonic numbers either p − 1 or p − 2 for every vertex is characterized. It is shown that for positive integers a, b and n ≥ 2 with 2 ≤ a ≤ b, there exists a connected graph G with rad m G = a, diam m G = b and m x (G) = n for some vertex x in G. Also, it is shown that for each triple m, n and p of integers with 1 ≤ n ≤ p − m − 1 and m ≥ 3, there is a connected graph G of order p, monophonic diameter m and m x (G) = n for some vertex x of G.