We consider unsupervised estimation of mixtures of discrete graphical models, where the class variable corresponding to the mixture components is hidden and each mixture component over the observed variables can have a potentially different Markov graph structure and parameters. We propose a novel approach for estimating the mixture components, and our output is a tree-mixture model which serve...

We consider an edge-isoperimetric problem (EIP) on the cartesian powers of graphs. One of our objectives is to extend the list of graphs for whose cartesian powers the lexicographic order provides nested solutions for the EIP. We present several new classes of such graphs that include as special cases all presently known graphs with this property. Our new results are applied to derive best poss...

We introduce nite relational structures called sketches, that represent edge crossings in drawings of nite graphs. We consider the problem of characterizing sketches in Monadic Second-Order logic. We answer positively the question for framed sketches, i.e., for those representing drawings of graphs consisting of a planar connected spanning subgraph (the frame) augmented with additional edges th...

Let c : E(G) → [k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̄(v) = (a1, . . . , ak), where ai is the number of edges incident to v with color i. Reorder c̄(v) for every v in G in nonincreasing order to obtain c∗(v), the color-blind partition of v. When c∗ induces a proper vertex coloring, that is, c∗(u) 6= c∗(v) for every edge uv in G, we say that c is col...

For a simple, undirected graph G with, at most one isolated vertex and no edges, labeling f:E(G)→{1,2,…,k1} of positive integers to the edges is called irregular if weights each has different value. The integer k1 then irregularity strength G. If number vertices in or order |G|, μ:E(G)→{1,2,…,k2} modular remainder divided by |G| k2 disjoint union two more graphs, denoted ‘+’, an operation where...

Blbzsik, Z., M. Hujter, A. Pluhir and Z. Tuza, Graphs with no induced C4 and 2K,, Discrete Mathematics 115 (1993) 51-55. We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class B of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members...

Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph G = (V,E), a geodesic triangle △(x, y, z) with x, y, z ∈ V is the union P (x, y)∪P (x, z)∪P (y, z) of three shortest paths connecting these vertices. A geodesic triangle △(x, y, z) is called δ-slim if for any vertex u ∈ V on any side P (x, y) the distance from u to P (x, z)∪P (y, z) is at most δ, i.e. ...

The atom graph of a graph is the graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all possible atom trees of this graph. We provide two efficient algorithms for computing this atom graph, with a complexity in O(min(n log n, nm,n(n+m)) time, which is no more than the complexity of computing the atoms in the gene...

All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For two graphs G, H, let N(G, H) denote the number of subgraphs of G isomorphic to H. Define also, for l >=0, N(I,H)= max N(G, H), where the maximum is taken over all graphs G with l edges. We determine N(l, H) precisely for all l -> 0 when H is a disjoint union of two stars, and also when H...

Let G = (V,E) be a graph of order n and size e. An (a, d)-vertexantimagic total labeling is a bijection α from V (G) ∪ E(G) onto the set of consecutive integers {1, 2, . . . , n + e}, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d. The vertex-weight of a vertex x is the sum of values α(xy) assigned to all edges xy incident to the ...