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In this paper, we focus on partial geometric designs with block sizes three and four. First, we show that a partial geometric design with block size three is a 2‐design, or a transversal design, or each line of a generalized quadrangle that is repeated λ times. Second, we introduce the concept of type of partial geometric designs with block size four and characterize partial geometric designs of type [λ,1,1] for some integer λ≥1.

Let [Formula: see text] be a finite undirected simple connected graph with vertex set [Formula: see text] and edge set [Formula: see text]. A vertex [Formula: see text] resolves two elements (vertices or edges) [Formula: see text] if [Formula: see text]. A subset [Formula: see text] of vertices in [Formula: see text] is called a mixed metric generator for [Formula: see text] if every two distinct elements (vertices and edges) of [Formula: see text] are resolved by some vertices of [Formula: see text]. The minimum cardinality of a mixed metric generator for [Formula: see text] is called the mixed metric dimension and is denoted by [Formula: see text]. In this paper, we study the mixed metric dimension for the plane graph of web graph [Formula: see text] and convex polytope [Formula: see text].

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