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In this paper 1 we consider higher isoperimetric numbers of a (finite directed) graph. In this regard we focus on the nth mean isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set of n disjoint subsets of the vertex set of the graph. We show that the second mean isoperimetric constant in this general setting, coincides with (the mean version of) the classical Cheeger constant of the graph, while for the rest of the spectrum we show that there is a fundamental difference between the nth isoperimetric constant and the number obtained by taking the minimum over all n-partitions. In this direction, we show that our definition is the correct one in the sense that it satisfies a Federer-Fleming-type theorem, and we also define and present examples for the concept of a supergeometric graph as a graph whose mean isoperimetric constants are attained on partitions at all levels. Moreover, considering the NP-completeness of the isoperimetric problem on graphs, we address ourselves to the approximation problem where we prove general spectral inequalities that give rise to a general Cheeger-type inequality as well. On the other hand, we also consider some algorithmic aspects of the problem where we show connections to orthogonal representations of graphs and following J. Malik and J. Shi (2000) we study the close relationships to the well-known k-means algorithm and normalized cuts method.

This paper is aimed to investigate some computational aspects of different isoperimetric problems on weighted trees. In this regard, we consider different connectivity parameters called minimum normalized cuts/isoperimteric numbers defined through taking minimum of the maximum or the mean of the normalized outgoing flows from a set of subdomains of vertices, where these subdomains constitute a partition/subpartition. We show that the decision problem for the case of taking k-partitions and the maximum (called the max normalized cut problem NCP M ) as well as the other two decision problems for the mean version (referred to as IPP m and NCP m ) are N P -complete problems. On the other hand, we show that the decision problem for the case of taking k-subpartitions and the maximum (called the max isoperimetric problem IPP M ) can be solved in linear time for any weighted tree and any k ≥ 2. Based on this fact, we provide polynomial time O(k)-approximation algorithms for all different versions of kth isoperimetric numbers considered. Moreover, when the number of partitions/subpartitions, k, is a fixed constant, as an extension of a result of B. Mohar (1989) for the case k = 2 (usually referred to as the Cheeger constant), we prove that max and mean isoperimetric numbers of weighted trees as well as their max normalized cut can be computed in polynomial time. We also prove some hardness results for the case of simple unweighted graphs and trees.

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Abstract:A k−clique covering of a simple graph G is a collection of cliques of G covering all the edges of G such that each vertex is contained in at most k cliques. The smallest k for which G admits a k−clique covering is called the local clique cover number of G and is denoted by lcc(G). Local clique cover number can be viewed as the local counterpart of the clique cover number that is equal to the minimum total number of cliques covering all edges. In this article, several aspects of the local clique covering problem are studied and its relationships to other well-known problems are discussed. In particular, it is proved that the local clique cover number of every claw-free graph is at most c / log , where is the maximum degree of the graph and c is a constant. It is also shown that the bound is tight, up to a constant factor. Moreover, regarding a conjecture by Chen et al. (Clique covering the edges of a locally cobipartite graph, Discrete Math 219(1-3)(2000), 17-26), we prove that the clique cover number of every connected claw-free graph on n vertices with the minimum degree δ, is at most n + c δ 4/3 log