Abstract:
We introduce and study the weighted $r$-path ideal of a weighted graph $G_\omega$, which is a common generalization of Conca and De Negri's $r$-path ideal for unweighted graphs and Paulsen and Sather-Wagstaff's edge ideal of the weighted graph. Over a field, we explicitly describe primary decompositions of these ideals, and we characterize Cohen-Macaulayness of these ideals for trees (with arbitrary $r$) and complete graphs (for $r=2$).

Abstract:
We find a class of block graphs whose binomial edge ideals have minimal regularity. As a consequence, we characterize the trees whose binomial edge ideals have minimal regularity. Also, we show that the binomial edge ideal of a block graph has the same depth as its initial ideal.

Abstract:
We prove two recent conjectures on some upper bounds for the Castelnuovo-Mumford regularity of the binomial edge ideals of some different classes of graphs. We prove the conjecture of Matsuda and Murai for graphs which has a cut edge or a simplicial vertex, and hence for chordal graphs. We determine the regularity of the binomial edge ideal of the join of graphs in terms of the regularity of the original graphs, and consequently prove the conjecture of Matsuda and Murai for such a graph, and hence for complete $t$-partite graphs. We also generalize some results of Schenzel and Zafar about complete $t$-partite graphs. We also prove the conjecture due to the authors for a class of chordal graphs.

Abstract:
We consider the edge ideals of large classes of graphs with whiskers and for these ideals we prove that the arithmetical rank is equal to the big height. Then we extend these results to other classes of squarefree monomial ideals, generated in any degree, proving that the same equality holds.

Abstract:
In this paper, we introduced the polarization of Koszul cycles and use it to study the depth function of powers of edge ideals of whisker graphs.

Abstract:
We prove that the edge ideals of line and cyclic graphs and their quotient rings satisfy the Stanley conjecture. We compute the Stanley depth for the quotient ring of the edge ideal associated to a cycle graph of length $n$, given a precise formula for $n\equiv 0,2 \pmod{3}$ and tight bounds for $n\equiv 1 \pmod{3}$. Also, we give bounds for the Stanley depth of a quotient of two monomial ideals, in combinatorial terms.

Abstract:
Reconstruction closings have all properties of a physical flooding of a topographic surface. They are precious for simplifying gradient images or, filling unwanted catchment basins, on which a subsequent watershed transform extracts the targeted objects. Flooding a topographic surface may be modeled as flooding a node weighted graph (TG), with unweighted edges, the node weights representing the ground level. The progression of a flooding may also be modeled on the region adjacency graph (RAG) of a topographic surface. On a RAG each node represents a catchment basin and edges connect neighboring nodes. The edges are weighted by the altitude of the pass point between both adjacent regions. The graph is flooded from sources placed at the marker positions and each node is assigned to the source by which it has been flooded. The level of the flood is represented on the nodes on each type of graphs. The same flooding may thus be modeled on a TG or on a RAG. We characterize all valid floodings on both types of graphs, as they should verify the laws of hydrostatics. We then show that each flooding of a node weighted graph also is a flooding of an edge weighted graph with appropriate edge weights. The highest flooding under a ceiling function may be interpreted as the shortest distance to the root for the ultrametric flooding distance in an augmented graph. The ultrametric distance between two nodes is the minimal altitude of a flooding for which both nodes are flooded. This remark permits to flood edge or node weighted graphs by using shortest path algorithms. It appears that the collection of all lakes of a RAG has the structure of a dendrogram, on which the highest flooding under a ceiling function may be rapidly found.

Abstract:
We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step $i$ in the minimal graded free resolution where there exists a minimal generator of degree $>i+3$, show that at this step the highest degree of a minimal generator is $i+4$, and determine the value of the corresponding graded Betti number $\beta_{i,i+4}$ in terms of the combinatorics of the associated bipartite graph. The results can then be easily extended to the non-squarefree case through polarization. We also study a family of ideals of regularity 4 that play an important role in our main result and whose graded Betti numbers can be completely described through closed combinatorial formulas.

Abstract:
Watersheds have been defined both for node and edge weighted graphs. We show that they are identical: for each edge (resp.\ node) weighted graph exists a node (resp. edge) weighted graph with the same minima and catchment basin.

Abstract:
This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a natural linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal-dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem, respectively. These results match the currently known best results for purely edge-weighted graphs.