We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. also provide examples oriented graphs that have non-Cohen-Macaulay vertex-weighted ideals, while the ideal their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes, Toledo.

A graph [Formula: see text] is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function defined on vertices. Then text]well-covered weight. For every text], set functions such text]-well-covered vector space, denoted as WCW(G). Deciding whether an input co-NP-complete. Therefore, finding WCW(G) co-NP-hard. generating subgraph induced complete b...

In this paper we review the state of the art on graph privacy with special emphasis on applications to online social networks, and we consider some novel aspects which have not been greatly covered in the specialized literature on graph privacy. The following key considerations are covered: (i) choice of different operators to modify the graph; (ii) information loss based on the cost of graph o...

A graph G is a fractional ( , b k )-critical covered if − U [ ]-covered for every ⊆ V ) with | = which first defined by (Zhou, Xu and Sun, Inf. Process. Lett. 152 (2019) 105838). Furthermore, they derived degree condition to be graph. In this paper, we gain an independence number connectivity verify that ≥ max {2 +1)( +1)+4 bk +5/4 ,( +1) 2

A nontrivial connected graph is matching covered if each of its edges lies in a perfect matching. Two types of decompositions of matching covered graphs, namely ear decompositions and tight cut decompositions, have played key roles in the theory of these graphs. Any tight cut decomposition of a matching covered graph results in an essentially unique list of special matching covered graphs, call...

The inflation GI of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorphic to the complete graph Kd, and each edge (xi, xj) of G is replaced by an edge (u, v) in such a way that u ∈ Xi, v ∈ Xj , and two different edges of G are replaced by non-adjacent edges of GI . The total domination number γt(G) of a graph G...

Six hens pecked a key (Experiment 1) or pushed a door (Experiment 2) to obtain food reinforcement. In both experiments and as an analogue of price changes, the response requirements were varied in two ways: by increasing the number of responses required and by increasing the required force of each response. The two price manipulations (response number and response force) had different effects o...