kmax-domset 0.1.1

K-Max algorithm for computing dominating sets in undirected graphs

kmax-domset

K-Max algorithm for computing dominating sets in undirected graphs.

A dominating set D of a graph G = (V, E) is a subset of vertices such that every vertex in V either belongs to D or is adjacent to at least one vertex in D.

This package implements the K-Max algorithm, which uses Karci centrality — a measure that combines the node degree in the original graph, its degree in a K-max spanning tree, and the sizes of fundamental cut-sets — to greedily select the most structurally important nodes into the dominating set.

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Installation

pip install kmax-domset

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Quick Start

from kmax_domset import KMaxAlgorithm

# Adjacency matrix: 1 = edge, 0 = no edge
adj = [
    [0, 1, 0, 0, 1],
    [1, 0, 1, 0, 0],
    [0, 1, 0, 1, 0],
    [0, 0, 1, 0, 1],
    [1, 0, 0, 1, 0],
]

algo = KMaxAlgorithm(adj)
D = algo.dominating_set()
print("Dominating set:", sorted(D))   # e.g. [0, 2]

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Visualisation

algo.visualize(D)                        # spring layout (default)
algo.visualize(D, layout="circular")     # circular layout
algo.visualize(D, layout="kamada")       # Kamada-Kawai layout
algo.visualize(D, layout="spectral")     # spectral layout

Red nodes are in the dominating set; blue nodes are dominated neighbours.

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Building the adjacency matrix

Manual (small graphs)

n = 6
adj = [[0] * n for _ in range(n)]

edges = [(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 0), (0, 3)]
for u, v in edges:
    adj[u][v] = 1
    adj[v][u] = 1

From NetworkX

import networkx as nx
from kmax_domset import KMaxAlgorithm

G = nx.petersen_graph()
n = G.number_of_nodes()
adj = [[0] * n for _ in range(n)]
for u, v in G.edges():
    adj[u][v] = 1
    adj[v][u] = 1

D = KMaxAlgorithm(adj).dominating_set()
print("Dominating set:", sorted(D))

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API

KMaxAlgorithm(adj_matrix)

Parameter	Type	Description
adj_matrix	list[list[int]]	Square symmetric adjacency matrix (0/1).

dominating_set() → set[int]

Returns a set of node indices forming a dominating set.

visualize(D=None, layout="spring")

Parameter	Type	Description
D	set[int]	Dominating set to highlight. Auto-computed if None.
layout	str	"spring" · "circular" · "kamada" · "spectral"

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Algorithm Overview

1. Phase 1 — K-max spanning tree: Build a spanning tree (forest for disconnected graphs) by greedily selecting the highest-degree unvisited node at each step using a max-priority queue.

2. Phase 2 — Fundamental cut-sets: For each tree branch (edge), compute the size of its fundamental cut-set — the number of graph edges (branches + chords) whose removal disconnects the two components formed by removing that branch.

3. Karci centrality: For each node v:
   K(v) = deg_G(v) + deg_T(v) + Σ cut_size(e) for all tree branches e incident to v.

4. Phase 3 — Greedy selection: Repeatedly pick the undominated node with the highest Karci centrality and add it to the dominating set until all nodes are covered.

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Requirements

• Python ≥ 3.8
• networkx >= 2.6
• matplotlib >= 3.4

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