We call a graph matching-covered if every line belongs to a perfect matching. We study the technique of "ear-decompositions" of such graphs. We prove that a non-bipartite matchingcovered graph contains K~ or K2@Ka (the triangular prism). Using this result, we give new characterizations of those graphs whose matching and covering numbers are equal. We apply these results to the theory of r-criti...

The tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. We determine t2(K(n1, n2, . . . , nk)) of the complete k-partite graph K(n1, n2, . . . , nk). In particular, we prove that t2(K(n,m)) = (m − 2)/2n + 2, whe...

The independence polynomial of a graph G is the function i(G, x) = ∑k≥0 ik xk , where ik is the number of independent sets of vertices in G of cardinality k. We prove that real roots of independence polynomials are dense in (−∞, 0], while complex roots are dense in C, even when restricting to well covered or comparability graphs. Throughout, we exploit the fact that independence polynomials are...

We give a formula for the v-number of graded ideal that can be used to compute this number. Then, we show edge I(G) graph G, induced matching number G is an upper bound when very well-covered, or has simplicial partition, well-covered connected and contains neither four, nor five cycles. In all these cases, lower regularity ring G. classify cycle vertices are shedding gain insight into family W...

Studying parallelism and concurrency in the double-pushout approach to graph transformations is mainly based on the so-called triple-pushout condition. The categories of sets and of graphs satisfy this condition. If we, however, consider graph morphisms that may change the labels in a welldefined manner (structurally labeled graphs), the triple-pushout condition does no longer hold true. In thi...

Motivated by the problem of predicting the release kinetics of matrix tablets, we study random walks on the contact graph of a random sphere packing of spheres of two sizes. For a random walk on the unweighted graph that terminates in a specified target set, we compare the euclidean distance covered to the number of steps. We find a linear dependence of the former on the latter, with proportion...

We study the minimum connected sensor cover problem (MIN-CSC) and the budgeted connected sensor cover (Budgeted-CSC) problem, both motivated by important applications (e.g., reduce the communication cost among sensors) in wireless sensor networks. In both problems, we are given a set of sensors and a set of target points in the Euclidean plane. In MIN-CSC, our goal is to find a set of sensors o...

A heterochromatic tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most p vertex-disjoint heterochromatic trees. In this paper we determi...

The Berge-Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is in exactly two of the perfect matchings. If the Berge-Fulkerson Conjecture is true, then what can we say about the proportion of edges of a cubic bridgeless graph that can be covered by k of its perfect matchings? This is the question we address in this paper. W...