Search games are attractive for their correspondence with classical width parameters. For instance, the invisible search number (a.k.a. node search number) of a graph is equal to its pathwidth plus 1, and the visible search number of a graph is equal to its treewidth plus 1. The connected variants of these games ask for search strategies that are connected, i.e., at every step of the strategy, ...

A connected pseudocomplete n-coloring of a graph G is a (non-proper) n-coloring of the vertices of G such that each color class induces a connected subgraph and for each pair of color classes there is an edge with one end of each color; this can be viewed as a kind of “inverse image” of a clique minor. The connected pseudoachromatic index of a graph G is the largest n for which the line graph o...

We characterize the family of connected subgraphs in terms of linear matrix inequalities (LMI) with additional integrality constraints. We then show that convex relaxations of the integral LMI lead to parameterization of all weighted connected subgraphs. These developments allow for optimizing arbitrary graph functionals under connectivity constraints. For concreteness we consider the connected...

Überhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, certain specialisation of überhomology, related to dominating sets in To this end, we interpret überhomology as poset and investigate its functoriality properties. then show the Euler characteristic graph coincides with an evaluation connected domi...

In the graph searching game the opponents are a set of searchers and a fugitive in a graph. The searchers try to capture the fugitive by applying some sequence of moves that include placement, removal, or sliding of a searcher along an edge. The fugitive tries to avoid capture by moving along unguarded paths. The search number of a graph is the minimum number of searchers required to guarantee ...

‎let $d_{n,m}=big[frac{2n+1-sqrt{17+8(m-n)}}{2}big]$ and‎ ‎$e_{n,m}$ be the graph obtained from a path‎ ‎$p_{d_{n,m}+1}=v_0v_1 cdots v_{d_{n,m}}$ by joining each vertex of‎ ‎$k_{n-d_{n,m}-1}$ to $v_{d_{n,m}}$ and $v_{d_{n,m}-1}$‎, ‎and by‎ ‎joining $m-n+1-{n-d_{n,m}choose 2}$ vertices of $k_{n-d_{n,m}-1}$‎ ‎to $v_{d_{n,m}-2}$‎. ‎zhang‎, ‎liu and zhou [on the maximal eccentric‎ ‎connectivity ind...

The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph (V,EN ) where EN = {{a, b} | a 6= b , {x, a} ∈ E and {x, b} ∈ E for some x ∈ V }. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite. We present some results concerning the k-iterated neighborhood graph N(G) := N(N(. . . N(G))) of G. In particular...