Summary: We give two algorithms for listing all simplicial vertices of a graph running in time \(O(n^\alpha)\) and \(O(e^{2\alpha/(\alpha+1)})= O(e^{1.41})\), respectively, where \(n\) and \(e\) denote the number of vertices and edges in the graph and \(O(n^\alpha)\) is the time needed to perform a fast matrix multiplication. We present new algorithms for the recognition of diamond-free graphs \((O(n^\alpha+ e^{3/2}))\), claw-free graphs \((O(e^{(\alpha+1)/2})= O(e^{1.69}))\), and \(K_4\)-free graphs \((O(e^{(\alpha+1)/2})= O(e^{1.69}))\). Furthermore, we show that counting the number of \(K_4\)’s in a graph can be done in time \(O(e^{(\alpha+1)/2})\). For all other graphs on four vertices we can count within \(O(n^\alpha+ e^{1.69})\) time the number of occurrences as induced subgraph.